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Prepared by: M. S. KumarSwamy, TGT(Maths) Page - A -
MATHEMATICS
PROBLEM SOLVING ASSESSMENT
QUANTITATIVE APTITUDE
for
CLASS – IX and X
2015 – 16
IMPORTANT CONCEPTS TOPIC WISE ALONG WITH SOLVED EXAMPLES
Prepared by
M. S. KUMARSWAMY, TGT(MATHS) M. Sc. Gold Medallist (Elect.), B. Ed.
Kendriya Vidyalaya GaCHiBOWli
Prepared by: M. S. KumarSwamy, TGT(Maths) Page - B -
PREFACE
It gives me great pleasure in presenting the Quantitative Aptitude Study material
for Problem Based Assessment (PSA) for Class IX and X Mathematics for II-term. It is in
strictly according to the latest guidelines issued by CBSE. The Central Board of
Secondary Education (CBSE) is planning to initiate a Problem Solving Assessment
(PSA) test for students of classes IX & XI. To be introduced from the second term of the
2012-2013 academic session, the tests aim to assess the students in the areas of
quantitative reasoning, qualitative reasoning and language conventions.
I avail this opportunity to convey my sincere thanks to respected sir, Shri S. Vijay
Kumar, Director, KVS ZIET Gwalior, respected madam Smt. R. Kalavathi, Deputy
Commissioner, KVS RO Hyderabad, respected sir Shri Isampal, Deputy Commissioner, KVS
RO Bhopal, respected sir Shri P. V. Sairanga Rao, Deputy Commissioner, KVS RO Varanasi,
respected sir Shri P. Deva Kumar, Deputy Commissioner, KVS RO Ahmedabad, respected sir
Shri Y. Arun Kumar, Assistant Commissioner(Acad), KVS Headquarter, New Delhi, respected
sir Shri Sirimala Sambanna, Assistant Commissioner, KVS RO Hyderabad, respected sir Shri.
K. L. Nagaraju, Assistant Commissioner, KVS RO Bangalore, respected sir
Shri.Gangadharaiah, Assistant Commissioner, KVS RO Bangalore and respected Shri M.K.
Kulshreshtha, Assistant Commissioner, KVS RO Chandigarh for their blessings, motivation
and encouragement in bringing out this notes in such an excellent form.
I also extend my special thanks to respected sir Shri. P. S. Raju, Principal, KV
Gachibowli, respected Smt. Nirmala Kumari M., Principal, KV Mysore & respected Shri. M.
Vishwanatham, Principal, KV Raichur for their kind suggestions and motivation while
preparing this Question Bank.
I would like to place on record my thanks to respected sir Shri. P. K. Chandran,
Principal, presently working in KV Bambolim. I have started my career in KVS under his
guidance, suggestions and motivation.
Inspite of my best efforts to make this notes error free, some errors might have
gone unnoticed. I shall be grateful to the students and teacher if the same are brought to my
notice. You may send your valuable suggestions, feedback or queries through email to
[email protected] that would be verified by me and the corrections would be
incorporated in the next year Question Bank.
M. S. KUMARSWAMY
Prepared by: M. S. KumarSwamy, TGT(Maths) Page - C -
DEDICATED
TO
MY FATHER
LATE SHRI. M. S. MALLAYYA
Prepared by: M. S. KumarSwamy, TGT(Maths) Page - 1 -
NUMBERS
IMPORTANT FACTS AND FORMULAE
I. NUMERAL : In Hindu Arabic system, we use ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 called digits to represent any number. A group of digits, denoting a number is called a numeral. We represent a number, say 689745132 as shown below :
Ten Crores (108)
Crores (107)
Ten Lacs (Millions)
(106)
Lacs (105)
Ten Thousands
(104)
Thousands (103)
Hundreds (102)
Tens (101)
Units (100)
6 8 9 7 4 5 1 3 2 We read it as : 'Sixty-eight crores, ninety-seven lacs, forty-five thousand, one hundred and thirty-two'. II PLACE VALUE OR LOCAL VALUE OF A DIGIT IN A NUMERAL : In the above numeral : Place value of 2 is (2 x 1) = 2; Place value of 3 is (3 x 10) = 30; Place value of 1 is (1 x 100) = 100 and so on. Place value of 6 is 6 x 108 = 600000000 III.FACE VALUE : The face value of a digit in a numeral is the value of the digit itself at whatever place it may be. In the above numeral, the face value of 2 is 2; the face value of 3 is 3 and so on. IV PRIME NUMBERS : A number greater than 1 is called a prime number, if it has exactly two factors, namely 1 and the number itself. Prime numbers upto 100 are : 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. Prime numbers Greater than 100 : Let p be a given number greater than 100. To find out whether it is prime or not, we use the following method : Find a whole number nearly greater than the square root of p. Let k > p . Test whether p is divisible by any prime number less than k. If yes, then p is not prime. Otherwise, p is prime. e.g,,We have to find whether 191 is a prime number or not. Now, 14 > 191 . Prime numbers less than 14 are 2, 3, 5, 7, 11, 13. 191 is not divisible by any of them. So, 191 is a prime number. V.TESTS OF DIVISIBILITY 1. Divisibility By 2 : A number is divisible by 2, if its unit's digit is any of 0, 2, 4, 6,
8. Ex. 84932 is divisible by 2, while 65935 is not.
2. Divisibility By 3 : A number is divisible by 3, if the sum of its digits is divisible by 3. Ex.592482 is divisible by 3, since sum of its digits = (5 + 9 + 2 + 4 + 8 + 2) = 30, which is divisible by 3. But, 864329 is not divisible by 3, since sum of its digits =(8 + 6 + 4 + 3 + 2 + 9) = 32, which is not divisible by 3.
3. Divisibility By 4 : A number is divisible by 4, if the number formed by the last
two digits is divisible by 4. Ex. 892648 is divisible by 4, since the number formed
Prepared by: M. S. KumarSwamy, TGT(Maths) Page - 2 -
by the last two digits is 48, which is divisible by 4. But, 749282 is not divisible by 4, since the number formed by the last tv/o digits is 82, which is not divisible by 4.
4. Divisibility By 5 : A number is divisible by 5, if its unit's digit is either 0 or 5. Thus, 20820 and 50345 are divisible by 5, while 30934 and 40946 are not.
5. Divisibility By 6 : A number is divisible by 6, if it is divisible by both 2 and 3. Ex. The number 35256 is clearly divisible by 2. Sum of its digits = (3 + 5 + 2 + 5 + 6) = 21, which is divisible by 3. Thus, 35256 is divisible by 2 as well as 3. Hence, 35256 is divisible by 6.
6. Divisibility By 8 : A number is divisible by 8, if the number formed by the last three digits of the given number is divisible by 8. Ex. 953360 is divisible by 8, since the number formed by last three digits is 360, which is divisible by 8. But, 529418 is not divisible by 8, since the number formed by last three digits is 418, which is not divisible by 8.
7. Divisibility By 9 : A number is divisible by 9, if the sum of its digits is divisible
by 9. Ex. 60732 is divisible by 9, since sum of digits * (6 + 0 + 7 + 3 + 2) = 18, which is divisible by 9. But, 68956 is not divisible by 9, since sum of digits = (6 + 8 + 9 + 5 + 6) = 34, which is not divisible by 9.
8. Divisibility By 10 : A number is divisible by 10, if it ends with 0. Ex. 96410, 10480 are divisible by 10, while 96375 is not.
9. Divisibility By 11 : A number is divisible by 11, if the difference of the sum of its
digits at odd places and the sum of its digits at even places, is either 0 or a number divisible by 11. Ex. The number 4832718 is divisible by 11, since : (sum of digits at odd places) - (sum of digits at even places) (8 + 7 + 3 + 4) - (1 + 2 + 8) = 11, which is divisible by 11.
10. Divisibility By 12 ; A number is divisible by 12, if it is divisible by both 4 and 3.
Ex. Consider the number 34632. (i) The number formed by last two digits is 32, which is divisible by 4, (ii) Sum of digits = (3 + 4 + 6 + 3 + 2) = 18, which is divisible by 3. Thus, 34632 is divisible by 4 as well as 3. Hence, 34632 is divisible by 12.
11. Divisibility By 14 : A number is divisible by 14, if it is divisible by 2 as well as 7. 12. Divisibility By 15 : A number is divisible by 15, if it is divisible by both 3 and 5. 13. Divisibility By 16 : A number is divisible by 16, if the number formed by the
last4 digits is divisible by 16. Ex.7957536 is divisible by 16, since the number formed by the last four digits is 7536, which is divisible by 16.
14. Divisibility By 24 : A given number is divisible by 24, if it is divisible by both3
and 8.
15. Divisibility By 40 : A given number is divisible by 40, if it is divisible by both 5 and 8.
16. Divisibility By 80 : A given number is divisible by 80, if it is divisible by both 5
and 16. Note : If a number is divisible by p as well as q, where p and q are co-primes, then
Prepared by: M. S. KumarSwamy, TGT(Maths) Page - 3 -
the given number is divisible by pq. If p arid q are not co-primes, then the given number need not be divisible by pq, even when it is divisible by both p and q. Ex. 36 is divisible by both 4 and 6, but it is not divisible by (4x6) = 24, since 4 and 6 are not co-primes.
VI. BASIC FORMULAE 1. (a + b)2 = a2 + b2 + 2ab 2. (a - b)2 = a2 + b2 - 2ab 3. (a + b)2 - (a - b)2 = 4ab 4. (a + b)2 + (a - b)2 = 2 (a2 + b2) 5. (a2 - b2) = (a + b) (a - b) 6. (a + b + c)2 = a2 + b2 + c2 + 2 (ab + bc + ca) 7. (a3 + b3) = (a +b) (a2 - ab + b2) 8. (a3 - b3) = (a - b) (a2 + ab + b2) 9. (a3 + b3 + c3 -3abc) = (a + b + c) (a2 + b2 + c2 - ab - bc - ca) 10. If a + b + c = 0, then a3 + b3 + c3 = 3abc. VII. DIVISION ALGORITHM OR EUCLIDEAN ALGORITHM If we divide a given number by another number, then :
Dividend = (Divisor x Quotient) + Remainder VIII. {i) (xn - an ) is divisible by (x - a) for all values of n. (ii) (xn - an) is divisible by (x + a) for all even values of n. (iii) (xn + an) is divisible by (x + a) for all odd values of n. IX. PROGRESSION A succession of numbers formed and arranged in a definite order according to certain definite rule, is called a progression. Arithmetic Progression (A.P.) : If each term of a progression differs from its preceding term by a constant, then such a progression is called an arithmetical progression. This constant difference is called the common difference of the A.P. An A.P. with first term a and common difference d is given by a, (a + d), (a + 2d),(a + 3d),..... The nth term of this A.P. is given by Tn =a + (n – 1) d. The sum of n terms of this A.P.
2 ( 1) ( ) ( )2 2 2n nn n nS a n d a a a l
SOME IMPORTANT RULES:
Rule 1: Sum of first n natural numbers : ( 1)2
n n
Rule 2: Sum of first n odd numbers : n x n = n2. Rule 3: Sum of first n even numbers = n(n+1)
Rule 4: Sum of squares of first n natural numbers : ( 1)(2 1)6
n n n
Rule 5: Sum of cubes of first n natural numbers : 2( 1)
2n n
Rule 6: If n is the number of numbers and n is even then n/2 numbers will be even and n/2 numbers will be odd among first n natural numbers. Rule 7: If n is odd , then there are (n+1)/2 odd numbers and (n-1)/2 even numbers Rule 8: The difference between the squares of two consecutive numbers is always
Prepared by: M. S. KumarSwamy, TGT(Maths) Page - 4 -
an odd number. Ex. 92 – 82 = 17 Rule 9: The difference between the squares of two consecutive numbers is the sum of the two consecutive numbers Ex. Find out the number of all even numbers from 1 to 300 ? Solution : Since 300 is an even number so total number of even numbers wil be (n/20) = (300/2) = 150 even numbers Ex. What is the sum of all the even numbers from 1 to 381 Solution : Even numbers will be = (381-1)/2= 190 Sum of even numbers = n * (n+1) = 190 (190+1) = 36,290 Ex. Find out of sum of all the odd numbers from 50 to 200 Solution : Required Sum = Sum of all odd numbers from 1 to 200 - Sum of all odd numbers from 1 to 50 : 1002 – 252 = 9375 Rule 10 : Dividend = (Divisor * Quotient) + Remainder Ex. What least number must be added to 7963 to make it exactly divisible by 65 ? Solution :On dividing 7963 by 65 we get 33 as remainder, So the number to be added will be 65 - 33 = 32 Ex. What least number must be subtracted from 7963 to make it exactely divisible by 65 ? Solution : On dividing 7963 by 65 we get 33 as remainder, So the number to be subtracted will be 33 Ex. Find the least number of five digits which is exactly divisible by 73 ? Solution :Least number of five digits will be 10000, on dividing 10000 by 73 we get 72 as remainder, so the number will be = 10000 + 72 = 10072 Rule : for finding least number add the remainder to the least number Ex. find the greatest number of five digits which is exactly divisible by 147 ? Solution : The greatest number of five digit will be 99999, on dividing it by 147 we get 39 as remainder, so the required number will be 99999 - 39 = 99960 Rule : For finding greatest number substract the remainder to the greatest number H.C.F. AND L.C.M. OF NUMBERS IMPORTANT FACTS AND FORMULAE I. Factors and Multiples : If a number a divides another number b exactly, we say that a is a factor of b. In this case, b is called a multiple of a.
II. Highest Common Factor (H.C.F.) or Greatest Common Measure (G.C.M.) or Greatest Common Divisor (G.C.D.): The H.C.F. of two or more than two numbers is the greatest number that divides each of them exactly.
There are two methods of finding the H.C.F. of a given set of numbers : 1. Factorization Method : Express each one of the given numbers as the product of prime factors.The product of least powers of common prime factors gives H.C.F.
Prepared by: M. S. KumarSwamy, TGT(Maths) Page - 5 -
2. Division Method: Suppose we have to find the H.C.F. of two given numbers. Divide the larger number by the smaller one. Now, divide the divisor by the remainder. Repeat the process of dividing the preceding number by the remainder last obtained till zero is obtained as remainder. The last divisor is the required H.C.F. Finding the H.C.F. of more than two numbers : Suppose we have to find the H.C.F. of three numbers. Then, H.C.F. of [(H.C.F. of any two) and (the third number)] gives the H.C.F. of three given numbers. Similarly, the H.C.F. of more than three numbers may be obtained.
III. Least Common Multiple (L.C.M.) : The least number which is exactly divisible by each one of the given numbers is called their L.C.M.
1. Factorization Method of Finding L.C.M.: Resolve each one of the given numbers into a product of prime factors. Then, L.C.M. is the product of highest powers of all the factors, 2. Common Division Method {Short-cut Method) of Finding L.C.M.: Arrange the given numbers in a row in any order. Divide by a number which divides exactly at least two of the given numbers and carry forward the numbers which are not divisible. Repeat the above process till no two of the numbers are divisible by the same number except 1. The product of the divisors and the undivided numbers is the required L.C.M. of the given numbers,
IV. Product of two numbers =Product of their H.C.F. and L.C.M. V. Co-primes: Two numbers are said to be co-primes if their H.C.F. is 1. VI. H.C.F. and L.C.M. of Fractions: 1.H C F= H.C.F. of Numerators 2.L C M = L.C.M of Numerators__ L.C.M. of Denominators H.C.F. of Denominators VII. H.C.F. and L.C.M. of Decimal Fractions: In given numbers, make the same number of decimal places by annexing zeros in some numbers, if necessary. Considering these numbers without decimal point, find H.C.F. or L.C.M. as the case may be. Now, in the result, mark off as many decimal places as are there in each of the given numbers.
VIII. Comparison of Fractions: Find the L.C.M. of the denominators of the given fractions. Convert each of the fractions into an equivalent fraction with L.C.M. as the denominator, by multiplying both the numerator and denominator by the same number. The resultant fraction with the greatest numerator is the greatest. MCQ QUESTIONS FOR PRACTICE 1. Find the least number which when divided by 6,15 and 18 leave remainder 5 in
each case (a) 90 (b) 180 (c) 95 (d)185 2. Divisibility by 2,5,10 can be checked by
(a) sum of digits (b) last digit (c) last two digits (d) last three digits
3. Which is greatest 3-digit number exactly divisible by 8,10,12
(a) 120 (b) 360 (c) 960 (d) 980
Prepared by: M. S. KumarSwamy, TGT(Maths) Page - 6 -
4. 4= 2x2 , 15= 3x5, so H.C.F. of 4 and 15 is (a) 0 (b) 1 (c) 2 (d) 3
5. Find the least number which when divided by 12, 16, 24 and 36 leaves a remainder 7 in each case. (a) 150 (b) 151 (c)144 (d) none of these
6. Renu purchases two bags of fertiliser of weights 75 kg and 69 kg. Find the
maximum value of weight which can measure the weight of the fertiliser exact number of times. (a) 150 (b) 138 (c)144 (d) none of these
7. Which of the following is divisible by 3?
(a)15287 (b) 15267 (c) 15286 (d) 152638
8. Which of the following is divisible by 9? (a)15287 (b) 15267 (c) 15286 (d) 152638
9. If a number is divisible by 9, it must be divisible by __.
(a) 6 (b) 3 (c) 2 (d) 12
10. The HCF of smallest composite number and the smallest prime number is (a) 0 (b) 1 (c) 2 (d) 3
11. Given that HCF(1152, 1664) = 128 the LCM(1152, 1664) is
(a) 14976 (b) 1664 (c) 1152 (d) none of these
12. The HCF of two numbers is 23 and their LCM is 1449. If one of the numbers is 161, then the other number is
(a) 23 (b) 207 (c) 1449 (d) none of these
13. Which one of the following rational number is a non-terminating decimal expansion:
(a) 3350
(b) 66180
(c) 615
(d) 411000
14. What is the H.C.F. of two consecutive even numbers
(a) 1 (b)2 (c) 4 (d) 8
15. What is the H.C.F. of two consecutive odd numbers (a) 1 (b) 2 (c) 4 (d) 8
16. The missing number in the following factor tree is (a) 2 (b) 6 (c) 3 (d) 9
17. If the HCF of 65 and 117 is expressible in the form 65m – 117, then the value of m is (a) 4 (b) 2 (c) 1 (d) 3
Prepared by: M. S. KumarSwamy, TGT(Maths) Page - 7 -
18. The largest number which divides 70 and 125, leaving remainders 5 and 8, respectively, is (a) 13 (b) 65 (c) 875 (d) 1750
19. If two positive integers a and b are written as a = x3y2 and b = xy3 ; x, y are prime numbers, then HCF (a, b) is (a) xy (b) xy2 (c) x3y3 (d) x2y2
20. If two positive integers p and q can be expressed as p = ab2 and q = a3b; a, b being
prime numbers, then LCM (p, q) is (a) ab (b) a2b2 (c) a3b2 (d) a3b3
21. HCF of 256, 320 is
a) 32 b) 16 c) 64 d) 4 22. If X = 25 × 3, Y = 24 × 3 × 11, Z = 24 × 32 × 11 then the G.C.D of X, Y and Z is
a) 24 × 3 b) 25 × 3 c) 24 × 32 d) 25 × 32 23. Observe the process and find the numbers x, y, z
a) x = 1, y = 2, z = 14 b) x = 1, y = 1,z = 14 c) x = 2, y = 1, z = 14 d) x = 2, y = 2, z = 14
24. Find the largest number which can exactly divide 513, 783 and 1107
a) 27 b) 26 c) 28 d) 29 25. Three drums contain 72 litres, 108 litres and 144 litres of oil. What biggest
measure can measure all the different quantities exactly a) 32 litres b) 38 litres c) 36 litres d) 34 litres
26. Find the smallest number exactly divisible by 12, 15, 20 and 27
a) 520 b) 560 c) 540 d) 580 27. LCM of 24, 54 and 72 is
a) 214 b) 218 c) 216 d) 224 28. If A = 37 × 48 × 53, B = 36 × 43 × 58 then LCM of A and B is
a) 37 × 48 × 53 b) 36 × 43 × 58 c) 37 × 48 × 58 d) None of these 29. Find the least number when divided by 6, 7, 8, 9 and 12 leaves the same
remainder 2 in each case a) 508 b) 504 c) 506 d) 502
30. HCF of three numbers is 12. If they be in the ratio 2 : 3 : 4 then the numbers
Prepared by: M. S. KumarSwamy, TGT(Maths) Page - 8 -
a) 24, 36, 48 b) 12, 18, 24 c) 10, 20, 30 d) None of these 31. The sum of the two numbers is 216 and their HCF is 27. The numbers are
a) 54, 162 b) 27, 189 c) 108, 108 d) None of these 32. Three different containers contain different quantities of mixtures of milk and
water, whose measurements are 384kg, 432kg and 480kg. What is the biggest measure must be there to measure all the different quantities exactly a) 48kg b) 46kg c) 42kg d) 44kg
33. The least number of square tiles required to pave the ceiling of a room 15m 60cm
and 9m 10 cm a) 86 b) 88 c) 84 d) 82
34. The largest number which divides 147, 252, 282 to leave the same remainder
a) 9 b) 15 c) 25 d) 35 35. The largest natural number which exactly divides the product of any consecutive
natural numbers is a) 6 b) 12 c) 24 d) 120
36. The least multiple of 7, which leaves a remainder 4, when divided by 6, 9, 15 and
18 is a) 94 b) 194 c) 364 d) 414
37. Six sirens commence tolling together and toll at intervals of 2, 4, 6, 8, 10 and 12
seconds respectively. In 40 minutes how many times, do they toll together. a) 21 times b) 22 times c) 23 times d) 24 times
38. The least number which when divided by 16, 18 and 21 leaves the remainders 3,
5, 8 respectively a) 996 b) 998 c) 995 d) 997
39. The HCF and LCM of two numbers are 44 and 264. If the first number is divided
by 2 the quotient is 44. The other number is a) 134 b) 136 c) 130 d) 132
40. What least number must be subtracted from 1294 so that the remainder when
divided by 9, 11, 13 will leave in each case the same remainder 6? a) 0 b) 1 c) 2 d) 3
41. The product of two numbers is 1200 and their HCF is 5. The LCM of the numbers
a) 220 b) 260 c) 240 d) 280 42. The HCF of two numbers is 12 and their difference is also 12. The numbers are
a) 66, 78 b) 70, 82 c) 94, 106 d) 96, 108 43. About the number of pairs which have 16 as their HCF and 126 as their LCM.,
We can definitely say that. a) Only one pair exists b) Only two pairs exist. c) Many such pairs exist d) No such pair exists
Prepared by: M. S. KumarSwamy, TGT(Maths) Page - 9 -
44. HCF of 1 3 5 7 9, , , ,2 4 6 8 10
is
a) 12
b) 110
c) 9120
d) 1120
45. What is the LCM of 2 4 5 10, , ,3 9 6 15
?
a) 103
b) 403
c) 203
d) 803
DECIMAL FRACTIONS IMPORTANT FACTS AND FORMULAE I. Decimal Fractions : Fractions in which denominators are powers of 10 are known as decimal fractions. Thus ,1/10=1 tenth=.1;1/100=1 hundredth =.01; 99/100=99 hundreths=.99;7/1000=7 thousandths=.007,etc II. Conversion of a Decimal Into Vulgar Fraction : Put 1 in the denominator under the decimal point and annex with it as many zeros as is the number of digits after the decimal point. Now, remove the decimal point and reduce the fraction to its lowest terms.
Thus, 25 1 2008 2510.25 ; 2.008100 4 1000 125
. 1.84 184 8 0.365 365;2.99 299 13 0.584 584
III. 1. Annexing zeros to the extreme right of a decimal fraction does not change its value. Thus, 0.8 = 0.80 = 0.800, etc.
2. If numerator and denominator of a fraction contain the same number of decimal places, then we remove the decimal sign.
Thus, 1.84 184 8 0.365 365;2.99 299 13 0.584 584
IV. Operations on Decimal Fractions : 1. Addition and Subtraction of Decimal Fractions : The given numbers
are so placed under each other that the decimal points lie in one column. The numbers so arranged can now be added or subtracted in the usual way.
2. Multiplication of a Decimal Fraction By a Power of 10 : Shift the decimal point to the right by as many places as is the power of 10.
Thus, 5.9632 x 100 = 596,32; 0.073 x 10000 = 0.0730 x 10000 = 730. 3. Multiplication of Decimal Fractions : Multiply the given numbers
considering them without the decimal point. Now, in the product, the decimal point is marked off to obtain as many places of decimal as is the sum of the number of decimal places in the given numbers. Suppose we have to find the product (.2 x .02 x .002). Now, 2x2x2 = 8. Sum of decimal places = (1 + 2 + 3) = 6. .2 x .02 x .002 = .000008.
4. Dividing a Decimal Fraction By a Counting Number : Divide the given number without considering the decimal point, by the given
counting number. Now, in the quotient, put the decimal point to give as many places of decimal as there are in the dividend. Suppose we have to find the quotient (0.0204 + 17). Now, 204 ^ 17 = 12. Dividend contains 4 places of decimal. So, 0.0204 + 17 = 0.0012.
5. Dividing a Decimal Fraction By a Decimal Fraction : Multiply both the dividend and the divisor by a suitable power of 10 to make divisor a whole number. Now, proceed as above.
Prepared by: M. S. KumarSwamy, TGT(Maths) Page - 10 -
Thus, 0.00066/0.11 = (0.00066*100)/(0.11*100) = (0.066/11) = .006V V. Comparison of Fractions : Suppose some fractions are to be arranged in ascending or descending order of magnitude. Then, convert each one of the given fractions in the decimal form, and arrange them accordingly.
Suppose, we have to arrange the fractions 3/5, 6/7 and 7/9 in descending order. now, 3/5=0.6,6/7 = 0.857,7/9 = 0.777.... since 0.857>0.777...>0.6, so 6/7>7/9>3/5 VI. Recurring Decimal : If in a decimal fraction, a figure or a set of figures is
repeated continuously, then such a number is called a recurring decimal. In a recurring decimal, if a single figure is repeated, then it is expressed by putting a dot on it. If a set of figures is repeated, it is expressed by putting a bar on the set Thus 1/3 = 0.3333….= 0.3; 22 /7 = 3.142857142857.....= 3.142857 Pure Recurring Decimal: A decimal fraction in which all the figures after the decimal point are repeated, is called a pure recurring decimal. Converting a Pure Recurring Decimal Into Vulgar Fraction : Write the repeated figures only once in the numerator and take as many nines in the denominator as is the number of repeating figures. thus ,0.5 = 5/9; 0.53 = 53/59 ;0.067 = 67/999;etc... Mixed Recurring Decimal: A decimal fraction in which some figures do not repeat and some of them are repeated, is called a mixed recurring decimal. e.g., 0.17333 = 0.173. Converting a Mixed Recurring Decimal Into Vulgar Fraction : In the numerator, take the difference between the number formed by all the digits after decimal point (taking repeated digits only once) and that formed by the digits which are not repeated, In the denominator, take the number formed by as many nines as there are repeating digits followed by as many zeros as is the number of non-repeating digits. Thus 0.16 = (16-1) / 90 = 15/19 = 1/6; 0.2273 (2273 22) / 9900 2251/ 9900
MCQ QUESTIONS FOR PRACTICE
1. What is the decimal expansion of 1000
9
a) 0.9 b) 9000 c) 0.009 d) 0.09
2. 20+9+104 +
1001 can be written in decimal as
a) 29.04 b) 29.40 c) 2940 d) 0.2940
3. The decimal form 107 +
1006 +
10004 can be written as
a) 76.40 b) 7.640 c) 0.764 d) 764.0
4. 700 +20 + 5 +100
9 can be written in decimal form as
a) 725.09 b) 725.9 c) 72.59 d) 7.259
5. 70 +2 + 100
9 can be written in decimal form as
a) 72.9 b) 729 c) 72.09 d) 7.209
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6. 230 610
can be written in decimal form as
a) 3062 b) 362 c) 36.2 d) none of these
7. 8600 210
can be written in decimal form as
a) 6002.8 b) 602.8 c) 628 d) none of these
8. 860 2100
can be written in decimal form as
a) 62.8 b) 62.008 c) 62.08 d) none of these 9. The sum of 0.007 + 8.5 +30.08 is
a) 38.587 b) 3.100 c) 18.508 d) 385.87 10. Lata spend Rs 9.50 for buying a pen and Rs 2.50 for one pencil .How much
money did she spend a) Rs 3.450 b) Rs 7 c) Rs 9.750 d) Rs 12
11. Find the value of 9.756 – 6.28
a)16.036 b)9.128 c)3.476 d)34.76 12. Find the value of 35 – 2.54
a)32.46 b)1.46 c)3.246 d)37.54 13. Subtract Rs. 18.25 from Rs. 20.75
a)Rs. 25 b) Rs. 39 c) Rs. 2.50 d)Rs. 3.9 14. Raju bought a book for Rs. 35.65. He gave Rs. 50 to the shopkeeper. How much
money did he get back from the shopkeeper? a)Rs. 36.15 b)Rs. 14.35 c)Rs. 80.65 d) Rs. 1.435
15. Akash bought vegetables weighing 10kg.Out of this 3kg 500g is onions, 2kg 75g
is tomatoes and the rest is potatoes. What is the weight of the potatoes? a)9.500kg b) 1.425kg c)5.575kg d)4.425kg
16. The number 0.125 can be written as fractions in lowest terms
a)81 b)
1000125 c)
20025 d)
405
17. 1mm = _____ a)0.1cm b)0.01 cm c) 1.0 cm d)0.001cm
18. Which one of the following is not true
a)1.431 < 1.490 b)3.3 > 3.300 c)0.3 < 0.4 d) 3 > 0.8 19. The length of a young gram plant is 65mm. its length in cm will be
a)6.5cm b)0.65cm c)0.065cm d)6.05 cm.
20. The length of Ramesh’s notebook is 9 cm 5 mm. What will be its length in cm? a)9.5cm b)0.95cm c)0.095cm d)9.05 cm.
21. Write 2.34 as fractions in lowest terms.
a) 1720
b) 1712500
c) 17220
d) none of these
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22. Write 0.342 as fractions in lowest terms.
a) 1712500
b) 171500
c) 1720
d) none of these
23. 725 Paisa in rupees can be written as a) 72.5 b) 0.725 c) 7.25 d) 0.0725
24. 4.19 m in cm can be written as
a) 419cm b) 41.9cm c) 0.419cm d) 41.09cm 25. 8888m in Km can be written as
a) 88.88Km b) 888.8Km c) 8.888Km d) 8888Km 26. 22g in Kg can be written as
a) 2.2Kg b) 0.022Kg c) 2.002Kg d) 2.02Kg
27. 3 4 61000 500 60 910 1000 10000
is the expended form of
a) 1569.3406 b) 1569.346 c) 1569.3046 d) 1569.643 28. If 78.36 = 78.36xy then x and y ?
a) 1 each b) 0 each c) 6 each d) 3 each 29. If 15 + 2.094 + 3.62 = X then X is
a) 20.724 b) 20.704 c) 20.714 d) 20.734 30. If 3.7 + 4.96 + 13.578+ 12.1347 = Y then Y + 99 =
a) 134.3727 b) 135.3727 c) 133.3727 d) 132.3727 31. If X = 3.7 + 2.95, Y = 8.7 + 3.999, Z = 10 then X + Y + Z =
a) 28.349 b) 30.349 c) 29.349 d) 27.349 32. When a decimal is multiplied by some power of 10 then the decimal point is
shifted to the____by as many digits as there are zeros in the multiplier ? a) left b) right c) top d) below
33. If 2.3 × x = 46, 6.9 × y = 138 then x × y × 0.138 =
a) 55.2 b) 56.2 c) 57.2 d) 54.2 34. Match the following:
a 14 – 2.9 p 13.907
b 25 – 12.09 q 11.1
c 36 – 22.009 r 13.991
d 46 – 32.093 s 12.91 a) a – p, b – r, c – q, d – s b) a – p, b – r, c – q, d – s c) a – q, b – s, c – r, d – p d) a – p, b – r, c – q, d – s
35. If A = 201.4 – 132.68; B = 11.62 – 6.068 then A – B =
a) 63.178 b) 63.158 c) 63.168 d) 63.188
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36. Match the following:
a 98.7034 × 10 p 987034
b 98.7034 × 1000 q 9870.34
c 98.7034 × 100 r 98703.4
d 98.7034 × 10000 s 987.034 a) a – s, b – r, c – q, d – p b) a – p, b – r, c – q, d – s c) a – p, b – r, c – q, d – s d) a – p, b – r, c – q, d – s
37. What should be added to 78.3091 to get 93?
a) 14.6809 b) 14.6709 c) 14.6909 d) 14.7009 38. What should be subtracted from 82 to get 73.22?
a) 8.68 b) 8.98 c) 8.88 d) 8.78 39. What is the excess of 90 over 67.214?
a) 21.786 b) 22.786 c) 23.786 d) 24.786 40. If 6258 × 74 = 463092 then 6.258 × 7.4 =
a) 46.3092 b) 4.63092 c) 463.092 d) 4630.92 41. If 0.4325 × 15 = 6.4875 then 43.25 × 1.5 =
a) 64.875 b) 648.75 c) 6.4875 d) 0.64875 42. If 6.57 × 12 = 78.84 then 0.0657 × 0.12 =
a) 0.07884 b) 0.7884 c) 0.007884 d) 0.0007884 43. If 0.0760 × 0.89 = 0.06764 then 7.6 × 0.00089 =
a) 6.764 b) 0.06764 c) 0.6764 d) 0.006764 44. Match the following:
a 3.7 + 2.99 p 6.001
b 0.47 + 3.3 q 4.19
c 0.5 + 3.69 r 3.77
d 3.47 + 2.531 s 6.69 a) a – p, b – r, c – q, d – s b) a – s, b – p, c – q, d – r c) a – s, b – r, c – q, d – p d) a – r, b – q, c – q, d – s
45. If 48.47 × 30 = X, 29.07 × 400 = Y then X + Y =
a) 13072.1 b) 13082.1 c) 13062.1 d) 13092.1
46. If 0.0478100 1000 10000
x y z then 6
10 100 1000 10000x y z
a) 0.4786 b) 0.4768 c) 0.4687 d) 0.4687
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47. 7.6 3.11.9
a) 12.6 b) 12.8 c) 12.2 d) 12.4
48. 0.04 0.160.24 0.36
a) 126
b) 229
c) 227
d) 129
49. 22.5 0.047.5 0.125
a) 14
b) 16
c) 15
d) 110
50. 4.5 0.25 0.03 10.40.225
a) 4.658 b) 4.778 c) 4.688 d) 4.798
SIMPLIFICATION
IMPORTANT FACTS AND FORMULAE I. ’BODMAS’Rule: This rule depicts the correct sequence in which the operations are to be executed,so as to find out the value of a given expression. Here, ‘B’ stands for ’bracket’ ,’O’for ‘of’ , ‘D’ for’ division’ and ‘M’ for ‘multiplication’, ‘A’ for ‘addition’ and ‘S’ for ‘subtraction’. Thus, in simplifying an expression, first of all the brackets must be removed, strictly in the order(), {} and []. After removing the brackets, we must use the following operations strictly in the order: (1)of (2)division (3) multiplication (4)addition (5)subtraction. II. Modulus of a real number : Modulus of a real number a is defined as |a| ={a, if a>0 -a, if a<0 Thus, |5|=5 and |-5|=-(-5)=5. III. Virnaculum (or bar): When an expression contains Virnaculum, before applying the ‘BODMAS’ rule, we simplify the expression under the Virnaculum. MCQ QUESTIONS FOR PRACTICE 1. What are the coefficients of x in the expression 8 – x + y?
(a) 1 (b) –1 (c) 8 (d) none of these 2. What are the coefficients of y in the expression 4x – 3y?
(a) 4 (b) –3 (c) 3 (d) none of these
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3. What are the coefficients of y in the expression yz2 + 5? (a) 5 (b) z (c) z2 (d) none of these
4. Write the expression for the statement: the sum of three times x and 11
(a) x +3 +11 ( b) 3x + 11 (c) 3 + 11x (d) 3x – 11
5. Write an expression : Raju s father s age is 5 years more than 3 times Raju s age . If Raju s age is x years , then father’s age is (a) 3x+5 (b) 5-3x (c) 3x-5 (d) 15x
6. Identify the coefficient of x in expression 8 – x + y
(a ) 0 (b) 8 (c) –1 (d) 1 7. The number of terms in 4p2q – 3pq2+5 is
(a) 7 (b) 3 (c) 1 (d) 4 8. The expression for sum of numbers a and b subtracted from their product is
(a) a + b – ab (b) ab – a + b (c) ab – (a+b) (d) ab + a – b. 9. The sum of mn + 5 – 2 and mn+3 is
(a) 2mn + 3 (b) 6 ( c) 2mn + 8 (d) 2mn + 6. 10. What is the statement for the expression 3mn + 5
(a) 5 more than 31 of product of m and n
(b) number 5 added to product of number m and n (c) number 5 added to 3 times the product of m and n. (d) 5 more than 3 times the product of the numbers m and n
11. The constant term in the expression 1 + x2 + x is
(a) 1 (b) 2 (c) x (d) x2
12. The coefficient of y3 in the expression y – y3 + y2 is (a) 1 (b) y (c) –y3 (d) –1
13. The number of terms in the expression 1.2ab – 2.4 b + 3.6a is
(a) 1.2 (b) –2.4 (c) 3.6a (d) 3 14. What is the numerical coefficient of y2 in the expression 2x2y – 15xy2 +7y
(a) –15x (b) –15 (c) 2 (d) 7 15. The expression x + y – xy is
(a) Monomial (b) Binomial (c) Trinomial (d) Quadrinomial 16. The expression xyz is
(a) Monomial (b) Binomial (c) Trinomial (d) Zero polynomial 17. From the following expressions 10pq, 7p, 8q, -p2q2, -7pq, -23, ab,3a,b.The like
terms are (a) 3,7p (b) 10 pq, –7pq (c) ab,3a,b (d)10pq,7p,8q
18. From the following expressing 3ab,a2,b2,a,5ab, –2ab,2a2 the three terms are
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(a) 3ab,5ab, –2ab (b) a2,a,2a2 (c)3ab,a2,b2 (d)2a2,a2, a 19. Sum of 3m and 2n is
(a)5mn (b) 3m+2n (c) 5m (d) 5n 20. Sum of xy, x+y and y+ xy is
(a) 2xy +2x+y (b) 3xy+2y (c) 2xy +x+y (d) 2xy+x+2y 21. The value of 21b – 32 + 7b – 20b is
(a) 48b – 32 (b) –8b – 32 (c)8b – 32 (d)28b – 52 22. Subtract a – b from a + b the result is
(a) 2a + 2b (b) 2a (c) 2b (d) 2a – 2b 23. Subtracting –5y2 from y2 ,the result is
(a) –4y2 (b) 6y2 (c) 4 y2 (d) –6y2 24. The value of expression 5n – 2, when n= –2 is
(a) –12 (b) 8 (c) 1 (d) –8
25. The value of expression 7a – 4b for a = 3, b = 2 is (a) 13 (b) 7a – 6b (c) 21a – 8b (d) 29
26. When x = 0, y = –1, then the value of expression 2x + 2y is
(a) 4 (b) 0 (c) –2 (d) 2 27. Factors of the term 15x2 in the expression 15x2 –13x are
(a) 15, x, x (b) 15,-13 (c) 15x2, –13x (d)15 28. Factors of the terms –4pq2 in the expression 9p2q2 –4 pq2 are
(a) 9 p2q2 , –4 pq2 (b)9, –4 (c) –4,p,q,q (d) –4 29. If the length of each side of the equilateral triangle is l, then the perimeter of the
equilateral triangle is (a) 3l (b) 3+l (c) 3-l (d) l/3
30. Which of the following is monomial
(a) 2x +3 (b) 2x (c) 4x + 2y + 3 (d) 4y + 5x + z – 1 31. Which of the following is trinomial
(a) 2a + 6b – 1 (b) 1 (c) 5a – 7 (d) a + b + c – 3 32. Terms with factors y in the expression 8 + xy + xyz are
(a) xy, xyz (b) x, xz (c) 8, xy, xyz (d) y, xz 33. Identify the terms in the expression x + y + 1 which are not constant
(a) x,y,1 (b) x, y (c) x,1 (d) y,1
34. The value of expression 4x – 3 at x=2 is (a) –4 (b) 5 (c) 4 (d) 2
35. The value of expression 5n2 + 5n – 2 for n = –2 is
(a) 13 (b) 3 (c) 8 (d) 12
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36. The value of expression 2a2 + 2b2 – ab for a=2, b=1is
(a) 2 (b) 8 (c) 6 (d) 10 37. The value of x + 7 + 4(x – 5) for x=2
(a) –3 (b) 31 (c) 12 (d) 37 38. The value of expression 2a – 2b – 4 – 5 + a at a=1, b=-2
(a) 10 (b) –2 (c) 12 (d) –4 39. What must be subtracted from 2a + b to get 2a – b
(a) 2b (b) 4a (c) 0 (d) 4a+4b
40. What must be added to 3x + y to get 2x + 3y (a) 5x + 4y (b) –x + 2y (c) x – 2y (d) x + 2y
41. Subtract a + 2b from sum of a – b and 2a+b (a) 2a – 2b (b) 4a +2b (c) 2b (d) –2a +2b
42. On simplifying (a + b – 3) – (b – a +3) + (a – b + 3) the result is
(a) a – b + 3 (b) a – b – 3 (c) 3a – b – 3 (d) 3a+b+3 43. What should be value of ‘a’ if y2 + y – a equals to 3 for y=1
(a) –1 (b) –5 (c) 5 (d) 0 44. What is an expression for the statement: “p is multiplied by 16”
(a) 16p (b) p/16 (c) p+16 (d) p-16 45. The expression for the statement: “ y multiplied by 10 and then 7 added to
product”. (a) 10 + y + 7 (b) 7y + 10 (c) 10y + 7 (d) 10y
46. What is the statement for the expression 2y – 9
(a) 2y subtracted from 9 (b) 9 subtracted from y and multiplied by 2 (c) 9 subtracted from 9 (d) thrice of y minus 9
47. Give expression for: “ 5 times of ‘y’ to which 3 is added”
(a) y +15 (b) 5y + 3 (c) 5y
+ 3 (d) 3y +5
48. The equation for the statement: one forth of a number minus 4 gives 4.
(a) 4x – 4 = 4 (b) 4x
– 4 = 4 (c) 14
x – 4 = 4 (d) x – 4 = 14
49. The area of triangle is’ xy’ where’ x’ is length and ‘y’ is breadth. If the length of
rectangle is increased by 5 units and breadth is decreased by 3 units, the new area of rectangle will be (a) (x – y)(x + 3) (b) xy+15 (c) (x + 5)(y – 3) (d) xy + 5 – 3
50. Area of rectangle of length’ 3x’ and breadth ‘5y ‘is
(a) 3x + 5y (b) 15xy (c)15x (d)15y
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SQUARE ROOTS AND CUBE ROOTS
IMPORTANT FACTS AND FORMULAE
Square Root: If x2 = y, we say that the square root of y is x and we write, y x .
Thus, 4 2, 9 3, 196 14, . Cube Root: The cube root of a given number x is the number whose cube is x. We denote the cube root of x by 3 x . Thus, 33 3 38 2 2 2 2, 343 7 7 7 7 etc.
Note: 1. xy x y 2. x xy y
MCQ QUESTIONS FOR PRACTICE 1. Which is the smallest three-digit perfect square?
(a) 100 (b) 101 (c) 121 (d) 144
2. Which is the greatest three-digit perfect square? (a) 999 (b) 961 (c) 962 (d) 970
3. Which is the greatest 4-digit perfect square? (a) 9999 (b) 9990 (c) 9800 (d) 9801
4. Which is the smallest 4-digit perfect square? (a) 1024 (b) 1025 (c) 1000 (d) 1016
5. What will be the number of digits in the square root of 25600? (a) 3 (b) 2 (c) 5 (d) 4
6. What will be the number of digits in the square root of 1296? (a) 2 (b) 3 (c) 1 (d) 4
7. The square root of 12.25 is _______________ . (a) 3.5 (b) 2.5 (c) 35 (d) 25
8. What is the length of the side of a square whose area is 441 cm2 ? (a) 21 (b) 22 (c) 20 (d) 12
9. In a right angle triangle ABC , right angled at B, AB=6cm, BC=8cm ,then AC= ____ .
(a) 10 (b) 12 (c) 21 (d) 14
10. Which least number should be subtracted from 629 so as to get a perfect square ? (a) 4 (b) 5 (c) 6 (d) 3
11. The square root of 1.21 is (a) 1.1 (b) 11 (c) 21 (d) 2.1
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12. What is the smallest square number which is divisible by each of the numbers 6,9 and 15 ?
(a) 900 (b) 810 (c) 630 (d) 720
13. The square of 1.2 is (a) 144 (b) 1.44 (c) 14.4 (d) 2.4
14. The square root of 169 is
(a) -13 (b) 1.3 (c) -1.3 (d) 1310
15. What is the length of the diagonal of a rectangle having dimensions 3cm and 4cm?
(a) 5 (b) 7 (c) 1 (d) 4 16. What will be the length of third side of a right angled triangle whose hypotenuse
is 5cm and one of the side is 3 cm ? (a) 2 (b) 3 (c) 4 (d) 5
17. Which of the following is not a perfect square ? (a) 81 (b) 18 (c) 100 (d) 121
18. Which is the smallest square number that is divisible by each of the number 4,9 and10?
(a) 900 (b) 810 (c) 800 (d) 920
19. Which of the following is not a square number? ( a) 4 ( b ) 9 ( c) 16 ( d ) 24
20. The square of 23 is : ( a ) 529 ( b ) 526 ( c ) 46 ( d ) 429
21. The square of which of the following would be even number? (a ) 2826 ( b ) 7779 ( c ) 1057 ( d ) 131
22. The square of which of the following would be odd number? ( a) 431 ( b) 272 ( c) 1234 ( d ) 7928
23. Which of the following is a perfect square ? ( a ) 45 ( b) 81 ( c ) 18 ( d ) 54
24. What will be the “ one’s digit” in the square of 1234 ? ( a ) 6 ( b) 2 ( c ) 8 ( d ) 9
25. What will be the number of zeros in the square of 400 ? ( a) 5 ( b) 1 ( c ) 3 ( d ) 4
26. The perfect square number between 30 and 40 is : ( a) 35 ( b ) 39 ( c ) 36 ( d ) 32
27. Which of the following number would have digit 6 at units place ? ( a ) 192 ( b ) 242 ( c) 252 ( d) 132
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28. Which of the following number would have digit 5 at units place : ( a ) 952 ( b) 592 ( c ) 242 (d) 422
29. Which of the following number would have digit 1 at units place ? ( a ) 812 ( b ) 182 ( c ) 542 ( d ) 952
30. How many natural numbers lie between 92 and 102?
(a)15 (b) 19 (c) 18 (d) 17 31. How many non square numbers lie between 112and 122?
(a) 21 (b) 23 (c) 22 (d) 20
32. 25 can be express as the sum of first _________consecutive odd numbers . (a) (b) 4 (c) 6 (d) 3
33. How many numbers lie between square of 12 and 13? (a) 21 (b) 23 (c)22 (d)24
34. What will be the value of ‘ x’ in Pythagorean triplet (6,8, x)? (a) 5 (b) 7 (c)10 (d) 11
35. The square of -9 is (a) -81 (b) 81 (c) 18 (d) -18
36. The square root of 6400 is
(a) 80 (b) 81 (c) 32 (d) 23 37. By which smallest number 90 must be multiplied so as to make it a perfect square
? (a) 10 (b) 2 (c) 5 (d) 3
38. By which smallest number 48 must be divided so as to make it a perfect square ?
(a) 2 (b) 3 (c) 6 (d) 4 39. Which smallest number should be added to 80 so as to make it a perfect square ?
(a) 2 (b) 3 (c) 1 (d) 4 40. What could be the possible “one’s digit” of the square root of 625?
(a) 5 (b) 0 (c) 4 (d) 8 41. The Smallest number by which 12348 must be divided to obtain a perfect square
is (a) 3 (b) 5 (c) 4 (d) 7
42. 0.9 = ? (a) 3 (b) 0.3 (c) 0.03 (d) 0.33
43. 1.0816 = ? (a) 1.04 (b) 1.286 (c) 0.904 (d) 1.35
44. 288128
= ?
(a) 1214
(b) 3214
(c) 5214
(d) 9214
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45. 0.9 1.6 = ? (a) 0.12 (b) 1.2 (c) 0.75 (d) 12
46. Which is the smallest three-digit perfect cube?
(a) 125 (b) 343 (c) 729 (d) 512
47. Which is the greatest three-digit perfect cube? (a) 125 (b) 343 (c) 729 (d) 512
48. Which of the following is not a perfect cube ?
(a) 1 (b) 9 (c) 8 (d) 27
49. The cube of 4 is _______________ . (a) 12 (b) 8 (c) 4 (d) 64
50. The value of 53 is __________ . (a) 125 (b) 15 (c) 10 (d) 75
51. The cube of an even number is always ____________ . (a) odd number (b) even number (c) prime number (d) none of these
52. The cube of an odd number is always __________ .
(a) odd number (b) even number (c) prime number (d) none of these
53. Each prime factor appears _________ times in its cube? (a) 2 (b) 3 (c) 1 (d) 4
54. Which of the following is Hardy-Ramanujan Number ? (a) 1724 (b)1725 (c) 1727 (d) 1729
55. By which smallest natural number 392 must be multiplied so as to make the product a perfect cube ?
(a) 2 (b) 14 (c) 7 (d) 49
56. The smallest natural number by which 243 must be multiplied to make the product a perfect cube is __________ .
(a) 3 (b) 9 (c) 8 (d) 7
57. The smallest natural number by which 704 must be divided to obtain a perfect cube is
(a) 22 (b) 12 (c) 11 (d) 13
58. The smallest natural number by which 135 must be divided to obtain a perfect cube is
(a) 5 (b) 3 (c) 15 (d) 9
59. Which of the following is not a perfect cube ? (a) 216 (b) 343 (c) 125 (d) 108
60. The expansion of a3 is ___________ . (a) 3 × a (b) a+a+a (c) 3 × 3 × 3 (d) a × a × a
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61. What will be the unit digit of the cube of a number ending with 2 ? (a) 8 (b) 4 (c) 2 (d) 6
62. What will be the unit digit of the cube of a number ending with 4 ? (a) 4 (b) 6 (c) 2 (d) 8
63. What will be the unit digit of the cube of a number ending with 6 ? (a) 4 (b) 6 (c) 2 (d) 8
64. A cuboid has dimensions 5cm , 2cm, 5cm .How many such cuboid will be needed to form a cube ?
(a) 20 (b) 10 (c) 5 (d) 2
65. How many cuboids of dimensions 15cm, 30cm ,15cm will be needed to form a cube ?
(a) 15 (b) 4 (c) 30 (d) 5
66. 729 is the value of _______________ . (a) 83 (b) 93 (c) 63 (d) 43
67. Which of the following is a perfect cube ?
(a) 125 (b) 135 (c) 145 (d) 115
68. What is the volume of a cube whose edge is 2cm ? (a) 8 (b) 6 (c) 10 (d) 4
69. The symbol for cube root is __________ . (a) (b) 3 (c) 4 (d) none of these
70. The cube root of 512 is ________ . (a) 8 (b) 32 (c) 16 (d) 2
71. The value of 3 343 is ____ . (a) 8 (b) 7 (c) 6 (d) 3
72. Which of the following is true for any natural number n? (a) n2>n3 (b) n3>n2 (c) n2=n3 (d) none of these
73. If the volume of a cube is 125 cm3 then what would be the length of its side? (a) 25 (b) 5 (c) 4 (d) 15
74. What will be the unit digit of the cube root of a number ends with 8? (a) 2 (b) 8 (c) 4 (d) 6
75. What will be the unit digit of the cube root of a number ends with 2? (a) 2 (b) 8 (c) 4 (d) 6
76. What will be the unit digit of the cube root of a number ends with 3? (a) 3 (b) 7 (c) 5 (d) 2
77. What will be the unit digit of the cube root of a number ends with 7? (a) 3 (b) 7 (c) 6 (d) 5
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78. 9 is the cube root of __________ .
(a) 343 (b) 729 (c) 629 (d) 81
79. The number of digits in the cube root of a 6-digit number is _______ . (a) 3 (b) 2 (c) 4 (d) 6
80. How many digits will be there in the cube root of 46656 ? (a) 2 (b) 1 (c) 3 (d) 4
81. How many digits will be there in the cube root of 512 ? (a) 1 (b) 2 (c) 3 (d) 4
82. What will be the unit digit of 3 15625 ? (a) 5 (b) 0 (c) 3 (d) 4
83. How many zeros will be there in the cube root of 27000? (a) 3 (b) 0 (c) 1 (d) 2
84. How many zeros will be there in the cube root of 800? (a) 3 (b) 0 (c) 1 (d) cube root does not
exist
85. If 73=343 , then 3 343 = _________ . (a) 3 (b) 7 (c) 13 (d) 9
86. If 83=512 , then 3 512 = __________ . (a) 3 (b) 7 (c) 13 (d) 9
87. What will be the unit digit of 3 216 ? (a) 3 (b) 6 (c) 4 (d) 2
88. Find the one’s digit of the cube of 149
(a) 2 (b) 3 (c) 9 (d) none of these 89. Find the cube root of 8000.
(a) 20 (b) 200 (c) 40 (d) none of these 90. Find the cube root of 0.0027.
(a) 3 (b) 0.3 (c) 0.03 (d) none of these
AVERAGE
CONCEPT: Average of numbers = Sum of all the numbersTotal numbers
MCQ QUESTIONS FOR PRACTICE 1. Average of first five multiples of 3 is ?
a) 11 b) 15 c) 9 d) 13
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2. In Arun’s opinion, his weight is greater than 65 kg but less than 72 kg. His brother doest not agree with Arun and he thinks that Arun’s weight is greater than 60 kg but less than 70 kg. His mother’s view is that his weight cannot be greater than 68 kg. If all are them are correct in their estimation, what is the average of different probable weights of Arun a) 68 kg b) 67 kg c) 69 kg d) data inadequate
3. A library has an average of 510 visitors on Sundays and 240 on other days. The
average number of visitors per day in a month of 30 days beginning with a Sunday is: a) 250 b) 276 c) 280 d) 285
4. The average of 20 numbers is zero. Of them, at the most, how many may be
greater than zero? a) 1 b) 19 c) 10 d) 0
5. Ages of ‘A’ and ‘B’ are in the ratio of 2 : 3 respectively. Six years hence the ratio
of their ages will become 8 : 11 respectively. What is B’s present age? a) 27 yrs b) 28 yrs c) 18 yrs d) 25 yrs
6. The average weight of 16 boys in a class is 50.25 kg and that of the remaining 8
boys is 45.15 kg. Find the average weights of all the boys in the class. a) 48.55 kg b) 49.25 kg c) 48 kg d) 47.55 kg
7. Sumitra has an average of 56% on her first 7 examinations. How much she should
make on her eighth examination to obtain an average of 60% on 8 examinations? a) 92% b) 68% c) 88% d) 78%
8. The total age of A and B is 12 years more than that of total age of B and C. C is
how many years younger than A? a) 12 b) 25 c) 26 d) C is elder than A
9. The average weight of A, B and C is 45 kg. If the average weight of A and B be
40 kg and that of B and C be 43 kg, then the weight of B is: a) 20 kg b) 31 kg c) 17 kg d) 26 kg
10. A car owner buys petrol at Rs.7.50, Rs. 8 and Rs. 8.50 per litre for three
successive years. What approximately is the average cost per litre of petrol if he spends Rs. 4000 each year? a) Rs. 7.98 b) Rs. 8 c) Rs. 9 d) Rs. 8.50
11. The average weight of 8 persons increases by 2.5 kg when a new person comes in
place of one of them weighing 65 kg. What might be the weight of the new person? a) 80 kg b) 85 kg c) 82 kg d) 76.5 kg
12. If the average marks of three batches of 55, 60 and 45 students respectively is 50,
55, 60, then the average marks of all the students is: a) 55 b) 53.33 c) 54.68 d) none of these
13. The average of five results is 46 and that of the first four is 45. The fifth result is ?
a) 12.5 b) 10 c) 1 d) 50
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14. The average weight of 8 person’s increases by 2.5 kg when a new person comes in
place of one of them weighing 65 kg. What might be the weight of the new person? a) Data inadequate b) 85 kg c) 76 kg d) 76.5 kg
15. In a school with 600 students, the average age of the boys is 12 years and that of
the girls is 11 years. If the average of the school is 11 years 9 months, then the number of girls in the school is: ? a) 450 b) 250 c) 150 d) 350
16. A pupil’s marks were wrongly entered as 83 instead of 63. Due to that the average
marks for the class got increased by half (1/2). The number of pupils in the class is: a) 40 b) 20 c) 73 d) 10
17. A family consists of two grandparents, two parents and three grandchildren. The
average age of the grandparents is 67 years, that of the parents is 35 years and that of the grandchildren is 6 years. What is the average age of the family?
a) none of these b) 5317
yrs c) 1327
yrs d) 4287
yrs
18. In the first 10 overs of a cricket game, the run rate was only 3.2. What should be
the run rate in the remaining 40 overs to reach the target of 282 runs? a) 6.25 b) 6.5 c) 6.75 d) 7
19. A grocer has a sale of Rs. 6435, Rs. 6927, Rs. 6855, Rs. 7230 and Rs. 6562 for 5
consecutive months. How much sale must he have in the sixth month so that he gets an average sale of Rs. 6500? a) Rs. 6001 b) Rs.5991 c) Rs.4991 d) Rs.6991
20. The average monthly income of P and Q is Rs. 5050. The average monthly
income of Q and R is Rs. 6250 and the average monthly income of P and R is Rs. 5200. The monthly income of P is: a) 3500 b) 4500 c) 4000 d) 5000
21. The captain of a cricket team of 11 members is 26 years old and the wicket keeper
is 3 years older. If the ages of these two are excluded, the average age of the remaining players is one year less than the average age of the whole team. What is the average age of the team? a) 23 yrs b) 24 yrs c) 25 yrs d) none of these
22. The average age of husband, wife and their child 3 years ago was 27 years and
that of wife and the child 5 years ago was 20 years. The present age of the husband is: a) none of these b) 35 yrs c) 40 yrs d) 50 yrs
23. The arithmetic mean of the scores of a group of students in a test was 52. The brightest 20% of them secured a mean score of 80 and the dullest 25% a mean score of 31. The mean score of remaining 55% is: ? a) 45 b) 54.6 approx. c) 50 d) 51.4 approx.
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24. The average of six numbers is 30. if the average of first four is 25 and that of last three is 35, the fourth number is : ? a) 35 b) 30 c) 25 d) 40
25. If the average marks of three batches of 55, 60 and 45 students respectively is 50,
55 and 60, then the average marks of all the students is: ? a) none of these b) 54.68 c) 53.33 d) 55
26. The captain of a cricket team of 11 members is 26 years old and the wicket keeper
is 3 years older. If the ages of these two are excluded, the average age of the remaining players is one year less than the average age of the whole team. What is the average age of the team? a) 23 yrs b) 24 yrs c) 25 yrs d) none of these
27. The average age of 24 boys and the teacher is 15 years. When the teaher’s age is
excluded, the agerage decreases by 1. What is the age of the teacher ? a) 38 yrs b) 39 yrs c) 40 yrs d) none of these
28. In the first 10 overs of a cricket game, the run rate was only 3.2. What should be
the run rate in the remaining 40 overs to reach the target of 282 runs? a) 6.75 b) 6.25 c) 6.5 d) 7
29. Out of 9 persons, 8 persons spent Rs. 30 each for their meals. The ninth one spent
Rs. 20 more than the average expenditure of all the nine. The total money spent by all of them was: ? a) Rs. 260 b) Rs.290 c) Rs.292.50 d) Rs.400.50
30. The average age of students of a class is 15.8 years. The average age of boys in
the class is 16.4 years and that of the girls is 15.4 years. The ratio of the number of boys to the number of girls in the class is: ? a) 3 : 5 b) 1 : 2 c) 2 : 3 d) 3 : 4
31. The average score of a cricketer for ten matches is 38.9 runs. If the average for the
first six matches is 42. Then find the average for the last four matches? a) 33.25 b) 33.5 c) 35 d) 34.25
32. Average of all prime numbers between 30 to 50 ?
a) 37.8 b) 39 c) 37 d) 39.8 33. A car owner buys petrol at Rs 7.50, Rs. 8 and Rs. 8.50 per litre for three
successive years. What approximately is the average cost per litre of petrol if he spends Rs. 4000 each year? a) Rs. 8.50 b) Rs.7.98 c) Rs.8 d) Rs.9
34. A cricketer has a certain average for 9 innings. In the tenth inning He scores 100
runs. thereby increasing his average by 8 runs. His new average is : ? a) 24 runs b) 32 runs c) 20 runs d) 28 runs
35. After replaceing an old member by a new member, It was found that the average
age of five members of a club is the same as it was 3 years ago. What is the difference between the ages on the replaed and the new member ? a) 8 yrs b) 15 yrs c) 4 yrs d) 2 yrs
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36. Average of 10 matches is 32, How many runs one should should score to increase his average by 4 runs.? a) 76 b) 70 c) 78 d) 80
37. When a student weighing 45 kgs left a class, the average weight of the remaining
59 students increased by 200g. What is the average weight of the remaining 59 students ? a) 55 b) 56 c) 57 d) 58
38. Distance between two stations A and B is 778 km. A train covers the journey from
A to B at 84 km per hour and returns back to A with a uniform speed of 56km per hour. Find the average speed of the train during the whole journey? a) 67.2 km/hr b) 67.0 km/hr c) 69.0 km/hr d) 69.2 km/hr
39. The average of 11 observations is 60. If the average of first five observations is 58
and that of the last five is 56, then the sixth observation is : ? a) 90 b) 110 c) 85 d) 100
40. There are two sections A and B of a class, consisting of 36 and 44 students
respectively. If the average weight of sections A is 40 kg and that of section b is 35 kg. Find the average weight of the whole class? a) 39.25 b) 36.25 c) 37.25 d) 38.25
SURDS AND INDICES I IMPORTANT FACTS AND FORMULAE I
1. LAWS OF INDICES: (i) am x an = am + n (ii) am / an = am-n (iii) (am)n = amn (iv) (ab)n = anbn (v)
n n
n
a ab b
(vi) a0 = 1
(vii) 1nna
a
2. SURDS: Let a be a rational number and n be a positive integer such that a1/n = n a is irrational. Then n a is called a surd of order n. 3. LAWS OF SURDS:
1 / 2
1/
( )
( )
( ) .
( )
nn
n n n
nn
n
i a a
ii a a
iii ab a b
a aivb b
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( )
( )
( )
nn
m n mn
m n mn
v a a
vi a a
vii a a
MCQ QUESTIONS FOR PRACTICE
1. The value of 2 / 532 is : (a) 2 (b) 4 (c) 16 (d) 14
2. The value of 3/ 416 is : (a) 4 (b) 12 (c) 8 (d) 48
3. The value of 1
3125
is :
(a) 15
(b) 125
(c) 115
(d) 1125
4. The value of 1 / 2 1 / 411 11 is :
(a) 1/ 411 (b) 3/ 411 (c) 1/811 (d) 1/ 211
5. The value of 3 / 26 4 is :
(a) 196
(b) 164
(c) 512 (d) 1512
6. The value of 23125 is :
(a) 5 (b) 25 (c) 45 (d) 35
7. The value of 3/ 225 is : (a) 5 (b) 25 (c) 125 (d) 625
8. The value of 1264 is :
(a) 8 (b) 4 (c) 16 (d) 32
9. The value of 1532 is :
(a) 16 (b) 160 (c) 2 (d) 18
10. The value of 13125 is :
(a) 5 (b) 25 (c) 45 (d) 35
11. The value of 329 is :
(a) 18 (b) 27 (c) – 18 (d) 127
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12. For Positive real numbers a and b, which is not true? (a) ab a b (b) 2a b a b a b
(c) a ab b (d) a b a b a b
13. Out of the following, the irrational number is (a) 1.5 (b) 2.477 (c) 1.277 (d)
14. To rationalize the denominator of 1a b
,we multiply this by
(a) 1a b
(b) 1a b
(c) a ba b
(d) a ba b
15. The number of rational numbers between 3 and 5 is
(a) One (b) 3 (c) none (d) infinitely many
16. If we add two irrational numbers, the resulting number (a) is always an irrational number (b) is always a rational number (c) may be a rational or an irrational number (d) always an integer 17. The rationalizing factor of 7 2 3 is
(a) 7 2 3 (b) 7 2 3 (c) 5 2 3 (d) 4 2 3
18. If 1 0.1428577 , then 4
7 equals
(a) 0.428571 (b) 0.571428 (c) 0.857142 (d) 0.285718 19. The value of n for which n be a rational number is
(a) 2 (b) 4 (c) 3 (d) 5
20. 3 126 27
equals
(a) 12
(b) 2 (c) 3 (d) 13
21. 3 3 3 2 equals
(a) 9 5 2 6 (b) 9 6 (c) 3 2 (d) 9 3 2 3 3 6
22. The arrangement of 2, 5, 3 in ascending order is (a) 2, 3, 5 (b) 2, 5, 3 (c) 5, 3, 2 (d) 3, 2, 5
23. If m and n are two natural numbers and mn = 32, then nmn is (a) 52 (b) 53 (c) 510 (d) 512
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24. If 10 3.162 , then the value of 110
is
(a) 0.3162 (b) 3.162 (c) 31.62 (d) 316.2
25. If 6 5 23 16 4
4 9 3
x
, then the value of x is
(a) 2 (b) 4 (c) -2 (d) 6
PRACTICE QUESTIONS
1. Evaluate the following expressions:
3 1
18 36
138
256 343( ) ( ) 15625 ( )6561 1331
6561( ) ( )34365536
i ii iii
iv v
2. Simplify: 32 488 12
3. Simplify: 73 3 2 2
4. Simplify: 4 3 2 3 4 12( ) 2 ( ) 2. 2. 32i ii 5. Evaluate the following expressions:
12 1 4 343 3 3 2
152 2 26
625( ) ( )27 27 27 ( ) 6.2581
( ) 0.000064 ( ) 17 8
i ii iii
iv v
6. Simplify: 3
2( 4 )2 5 6
7. If 3 52
a then find the value of 2
2
1aa
.
8. Simplify: 1
3 41 13 3
4 6 34
( ) 5 8 27 ( ) 45 3 20 4 5
24 54( ) ( ) 12 7 ( ) 28 78 9
i ii
iii iv v
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9. Simplify:
23
3 54
2
1( ) 3 2 27 ( ) 3 53
( ) 81 8 216 15 32 225
3 1 3 3( ) ( )3 68 2
i ii
iii
iv v
10. Find the value of
2 3 13 4 5
4 1 2
216 256 243
11. Simplify:
1 13 2
1 26 3
9 27
3 3
12. Simplify:
24 12 61 33 3 3 2
2 1 111 1 21 3 34
3 3 321
3
3 8 32 1( ) 1 2 3 ( ) ( )5 5 5 27
8 16( ) 625 ( ) ( )64 64 6432
i ii iii
iv v vi
13. Find the value of a and b in each of the following:
3 2 3 7 7 5( ) 2 ( ) 7 ( ) 53 2 3 7 7 5
i a b ii a b iii a b
14. Simplify each of the following by rationalizing the denominator.
6 4 2 5 2 5 2( ) ( )6 4 2 5 2 5 2
i ii
15. If x = 3 23 2
, find (i) 22
1xx
(ii) 44
1xx
.
PERCENTAGE, PROFIT & LOSS & DISCOUNT IMPORTANT FACTS AND FORMULAE 1. Concept of Percentage : By a certain percent ,we mean that many hundredths.
Thus x percent means x hundredths, written as x%.
To express x% as a fraction : We have , x% = 100
x .
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Thus, 20% = 20 1100 5
; 48% = 48 12100 25
, etc.
To express a/b as a percent : We have, ab
= 100%ab .
Thus, 1 1 100% 25%4 4
2. If the price of a commodity increases by R%, then the reduction in consumption
so as not to increase the expenditure is 100%100
RR
If the price of the commodity decreases by R%,then the increase in consumption
so as to decrease the expenditure is 100%100
RR
3. Results on Population : Let the population of the town be P now and suppose it
increases at the rate of R% per annum, then :
1. Population after n years = 1100
nRP
2. Population n years ago = 1
100
nPR
4. Results on Depreciation : Let the present value of a machine be P. Suppose it
depreciates at the rate R% per annum. Then,
1. Value of the machine after n years = 1100
nRP
2. Value of the machine n years ago = 1
100
nPR
PROFIT AND LOSS
COST PRICE: The Price at which article is purchased. It is abbreviated as C.P.
SELLING PRICE: The price at which article is sold. It is abbreviated as S.P. PROFIT OR GAIN: If SP is greater than CP, then the selling price is said to have a profit or gain
LOSS: If SP is less than CP, then the selling price is said to have a loss.
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FORMULA 1.2.
3. % 100%
4. % 100%
100 %5.100
100 %6.1001007.
100 %1008.
100 %
Gain SP CPLoss CP SP
GainGainCP
LossLossCPGainSP CP
LossSP CP
CP SPGain
CP SPLoss
Discount = % . .100
Discount M P
S.P. = M.P. – Discount 100. .
100 %M P SP
Discount
MCQ QUESTIONS FOR PRACTICE 1. 72% of 25 students are good in hindi, how many are not good in hindi?
a) 16 b) 14 c) 18 d) 7
2. In a computer lab, there are 3 computers for every 6 students. How many computers will be needed for 24 students? a) 12 b) 14 c) 16 d) none of these
3. Out of 32 students, 8 are absent. What percent of the students are present? a) 75% b) 64% c) 60% d) none of these
4. There are 25 radios, 16 of them are out of order. What percent of radios are out of order? a) 75% b) 64% c) 60% d) none of these
5. A shop has 500 parts, out of which 5 are defective. What percent are not defective? a) 75% b) 99% c) 90% d) none of these
6. There are 120 voters, 90 of them voted yes. What percent voted yes? a) 75% b) 99% c) 90% d) none of these
7. The population of a town increased from 1,75,000 to 2,62,500 in a decade. The
average percent increase of population per year is: a) 7% b) 5% c) 9% d) 8.75%
8. A survey of 40 children showed that 25% liked playing football. How many children not liked playing football? a) 90 b) 60 c) 30 d) none of these
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9. When 15% is lost in grinding wheat, a country can export 30 lakh tons of wheat. On the other hand, if 10% is lost in grinding, it can export 40 lakh tons of wheat. The production of wheat in the country is: (a) 40 lakh tons (b) 400 lakh tons (c) 200 lakh tons (d) 900 lakh tons
10. In a History examination, the average for the entire class was 80 marks. If 10% of
the students scored 95 marks and 20% scored 90 marks, what was the average marks of the remaining students of the class? (a) 60 (b) 72 (c) 75 (d) 85
11. A’s marks in Biology are 20 less than 25% of the total marks obtained by him in
Biology, Maths and Drawing. If his marks in Drawing be 50,what are his marks in Maths? (a) 60 (b) 47 (c) 63 (d) cannot be determined
12. In an election between two candidates, one got 65%of the total valid votes, 20%
of the votes were invalid. If the total number of votes was 6500, the number of valid votes that the other candidate got, was (a) 1820 (b) 1840 (c) 1824 (d) 1844
13. 8% children of a class of 25 like getting wet in the rain. How many children do
not like getting wet in the rain. a) 20 b) 22 c) 23 d) none of these
14. Rahul bought a sweater and saved Rs 20 when a discount of 25% was given. What
was the price of the sweater before the discount? a) Rs 30 b) Rs 40 c) Rs 60 d) Rs 80
15. Out of 15,000 voters in a constituency, 60% voted. Find the number of voters who
did not vote. a) 9000 b) 6000 c) 3000 d) none of these
16. Meeta saves Rs 400 from her salary. If this is 10% of her salary. What is her
salary? a) 4000 b) 6000 c) 3000 d) none of these
17. A local cricket team played 20 matches in one season. It won 25% of them. How
many matches did they lose? a) 12 b) 14 c) 16 d) none of these
18. A school team won 6 games this year against 4 games won last year. What is the
per cent increase? a) 75% b) 50% c) 60% d) none of these
19. The number of illiterate persons in a country decreased from 150 lakhs to 100
lakhs in 10 years. What is the percentage of decrease?
a) 30% b) 50% c) 33 13
% d) none of these
20. Cost of an item is Rs 50. It was sold with a profit of 12%. Find the selling price.
a) Rs 56 b) Rs 60 c) Rs 70 d) none of these
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21. How much will an item cost if 10% discount is given on the marked price Rs 100
a) 90 b) 110 c) 95 d) 85
22. A football team won 10 matches out of the total number of matches they played. If their win percentage was 40, then how many matches did they play in all? a) 10 b) 25 c) 40 d) none of these
23. If Chameli had Rs 600 left after spending 75% of her money, how much did she
have in the beginning? a) Rs 3000 b) Rs 2400 c) Rs 2600 d) Rs 2800
24. The price of a scooter was Rs 34,000 last year. It has increased by 20% this year. What is the price now? a) Rs 40800 b) Rs 32300 c) Rs 40000 d) none of these
25. The price of a scooter was Rs 34,000 last year. It has decreased by 5% this year.
What is the price now? a) Rs 40800 b) Rs 32300 c) Rs 40000 d) none of these
26. Mohit bought a CD for Rs. 750 and sold it Rs. 875. Find his gain or loss percent.
a) 5% b) 16% c) 6% d) 16 23
%
27. Rahul purchased a table for Rs. Rs. 1260 and due to some scratches on its top he had to sell it for Rs. 1197. Find his loss or gain percent.
a) 5% b) 4% c) 6% d) 16 23
%
28. Raghu bought an almirah for Rs. 6250 and spent Rs. 375 on its repairs. Then he sold it for Rs. Rs. 6890. Find his gain or loss percent.
a) 5% b) 4% c) 6% d) 16 23
%
29. A vendor bought 20 oranges for Rs. 56 and sold them at Rs. 35 per dozen. Find
his gain or loss percent.
a) 5% b) 4 16
% c) 6% d) 16 23
%
30. The cost of a flower vase is Rs 120. If the shopkeeper sells it at a loss of 10%, find
the price at which it is sold. a) Rs 108 b) Rs 450 c) Rs 160 d) none of these
31. Selling price of a toy car is Rs 540. If the profit made by shopkeeper is 20%, what is the cost price of this toy? a) Rs 108 b) Rs 450 c) Rs 160 d) none of these
32. The marked price of a ceiling fan is Rs. 1250 and the shopkeeper allows a
discount of 6% on it. Find the selling price of the fan. a) Rs 1180 b) Rs 1175 c) Rs 1160 d) none of these
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33. A trader marks his goods at 40% above the cost price and allows a discount of 25%. What is his gain? a) Rs 118 b) Rs 175 c) Rs 105 d) none of these
34. A dealer purchased a washing machine for Rs. 7660. He allows a discount of 12% on its marked price and still gains 10%. Find the marked price of the machine. a) Rs 9800 b) Rs 9675 c) Rs 9575 d) none of these
35. On selling a fan for Rs. 810. Sunil gains 8%. For how much did he purchase it? a) Rs 780 b) Rs 750 c) Rs 760 d) none of these
36. On selling a table for Rs. 987. Ramesh loses 6%. For how much did he purchase
it? a) Rs 1500 b) Rs 105 c) Rs 1050 d) none of these
37. Rashmi buys a calculator for Rs. 720 and sells it at a loss of 6 23
%. For how much
does she sell it? a) Rs 700 b) Rs 650 c) Rs 672 d) none of these
38. The difference of two numbers is 20% of the larger number. If the smaller number
is 20, then the larger number is: (a) 25 (b) 46 (c) 27 (d) 82
39. When any number is divided by 12, then dividend becomes 1/4th of the other
number. By how much percent first number is greater than the second number? (a) 165 (b) 200 (c) 300 (d) 400
40. If one number is 80% of the other and 4 times the sum of their squares is 656, then
the numbers are: (a) 6,8 (b) 8, 10 (c) 16, 20 d) 10, 15
41. Two numbers A and B are such that the sum of 5% of A and 4% of B is two-third
of the sum of 6% of A and 8% of B. Find the ratio of A : B. (a) 1 : 2 (b) 3 : 1 (c) 3 : 4 (d) 4 : 3
42. Three candidates contested an election and received 1136, 7636 and 11628 votes
respectively. What percentage of the total votes did the winning candidate get? (a) 57% (b) 77% (c) 80% (d) 90%
43. The population of a town increased from 1,75,000 to 2,62,500 in a decade. The
average percent increase of population per year is: (a) 7% (b) 5% (c) 9% (d) 8.75%
44. A student multiplied a number by 3 instead of 5/3. What is the percentage error in
the calculation? (a) 36% (b) 44% (c) 55% (d) 35%
45. A tempo is insured to the extent of 4/5 of its original value. If the premium on it at
the rate of 1.3 percent amounts to Rs. 910, the original value of the tempo is: (a) Rs.78,000 (b) Rs.78,500 (c) Rs.80,000 (d) Rs.87,500
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RATIO AND PROPORTION
IMPORTANT FACTS AND FORMULAE
I. RATIO: The ratio of two quantities a and b in the same units, is the fraction a/b and we write it as a:b. In the ratio a:b, we call a as the first term or antecedent and b, the second term or consequent. Ex. The ratio 5: 9 represents 5/9 with antecedent = 5, consequent = 9. Rule: The multiplication or division of each term of a ratio by the same non-zero number does not affect the ratio. Ex. 4: 5 = 8: 10 = 12: 15 etc. Also, 4: 6 = 2: 3.
2. PROPORTION: The equality of two ratios is called proportion. If a: b = c: d, we write, a: b:: c : d and we say that a, b, c, d are in proportion . Here a and d are called extremes, while b and c are called mean terms. Product of means = Product of extremes. Thus, a: b:: c : d <=> (b x c) = (a x d). 3. (i) Fourth Proportional: If a : b = c: d, then d is called the fourth proportional to a, b, c. (ii) Third Proportional: If a: b = b: c, then c is called the third proportional to a and b. (iii) Mean Proportional: Mean proportional between a and b is square root of ab 4. (i) COMPARISON OF RATIOS: We say that (a: b) > (c: d) <=> (a/b)>(c /d). (ii) COMPOUNDED RATIO: The compounded ratio of the ratios (a: b), (c: d), (e : f) is (ace: bdf) 5. (i) Duplicate ratio of (a : b) is (a2 : b2). (ii) Sub-duplicate ratio of (a : b) is (√a : √b). (iii)Triplicate ratio of (a : b) is (a3 : b3). (iv) Sub-triplicate ratio of (a : b) is (a ⅓ : b ⅓ ). (v) If (a/b)=(c/d), then ((a+b)/(a-b))=((c+d)/(c-d)) (Componendo and dividendo)
6. VARIATION: (i) We say that x is directly proportional to y, if x = ky for some constant k and we write, x y. (ii) We say that x is inversely proportional to y, if xy = k for some constant k and we write, x∞(1/y)
PARTNERSHIP 1. Partnership: When two or more than two persons run a business jointly, they are
called partners and the deal is known as partnership. 2. Ratio of Division of Gains:
i) When investments of all the partners are for the same time, the gain or loss is distributed a among the partners in the ratio of their investments. Suppose A and B invest Rs. x and Rs. y respectively for a year in a business, then at
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the end of the year: (A’s share of profit) : (B's share of profit) = x : y.
ii) When investments are for different time periods, then equivalent capitals are calculated for a unit of time by taking (capital x number of units of time). Now, gain or loss is divided in the ratio of these capitals. Suppose A invests Rs. x for p months and B invests Rs. y for q months, then (A’s share of profit) : (B's share of profit) = xp : yq. 3. Working and Sleeping Partners: A partner who manages the business is known . as a working partner and the one who simply invests the money is a sleeping partner. MCQ QUESTIONS FOR PRACTICE 1. The ratio of 90 cm to 1.5 m is...........
(a) 3 : 5 (b) 5 : 3 (c) 60 : 1 (d) 4 : 3 2. 6 : 4 is equivalent ratio of ..............
(a) 2 : 3 (b) 3 : 2 (c) 1 : 2 (d) 1 : 4 3. Find the ratio of 81 to 108 ?
(a) 3 : 4 (b) 5 : 9 (c) 4 : 3 (d) 9 : 20
4. Fill in the blank :- 15 ......18 6
(a) 5 (b) 4 (c) 3 (d) 7 5. Find the value of x in 4 : 3 = x : 12 ?
(a) 4 (b) 12 (c) 16 (d) 3 6. In proportion first and the last terms are called ______________________ .
(a) Mean terms (b) Extreme terms (c) Middle terms (d) None of these 7. The ratio is said to be in simplest form if common factor is _____________.
(a) 1 (b) 0 (c) -1 (d) None of these 8. Three terms a , b , c are said to be in proportion if ______________ .
(a) a : b = b : c (b) a : b = c : b (c) b : a = c : a (d) c : a = a : b 9. Four terms a , b , c , d are said to be in proportion if ___________________ .
(a) a : b = c : d (b) a : c = d : b (c) a : d = b : c (d) None of these 10. If the cost of 6 cans of juice is Rs 210, then what is the cost of 4 cans of juice is
(a) Rs 120 (b) Rs 140 (c) Rs 100 (d) Rs 80 11. Fill in the blank :- 32 m : 64 m = 6 sec : ______
(a) 13 sec (b) 12 sec (c) 8 sec (d) 24sec 12. Which of the following is correct :-
(a) 3 : 4 = 15 : 25 (b) 12 : 24 = 6 : 12 (c) 7 : 3 = 14 : 3 (d) 5 : 10 = 9 : 20
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13. The ratio of 15 Kg to 75 Kg is........... (a) 1 : 5 (b) 5 : 1 (c) 3 : 5 (d) 15 : 3
14. 7 : 42 is equivalent ratio of ..............
(a) 7 : 6 (b) 6 : 1 (c) 1 : 6 (d) 6 : 7 15. Find the ratio of 33 Km to 121 Km ?
(a) 3 : 11 (b) 11 : 3 (c) 3 : 7 (d) 7 : 3 16. A stick 1.5m long casts a shadow 4.5m long at the same time when a pole casts a
shadow 9.6m long. Find the length of the pole a) 3.4m b) 3.6m c) 3.2m d) 3.8m
17. Find three numbers in the 2 : 3: 5, the sum of whose squares is 342
a) 6, 8, 9 b) 6, 9 , 15 c) 4, 6, 10 d) 10, 15, 25 18. Three utensils contain equal mixtures of alcohol and water in the ratio 5 : 2, 3 : 1
and 6 : 1 respectively. If all the solutions are mixed together find the ratio of milk and water in the final mixture. a) 19 : 165 b) 65 : 19 c) 63 : 19 d) 67 : 19
19. Three friends divide Rs 1110/– among themselves in the 1 1 1, ,4 5 6
. Find the largest
share? a) Rs 460 b) Rs 470 c) Rs 450 d) Rs 490
20. The boys and girls in a school are into ratio 9 : 5. If the number of girls are 225
then the number of ways are a) 415 b) 400 c) 405 d) 425
21. An alloy contains copper and zin (in the ration 9 : 5). If difference in the quantities
of two metals is 4000gms. What is the weight of the alloy in Quintals a) 1.4 b) 1.6 c) 1.2 d) 1.8
22. The ratio between the rates of walking of X and Y is 3 : 4. if the time taken by Y
to cover a certain distance is 24 minutes, how much time is taken by X to cover the same distance? a) 36mts b) 40mts c) 32mts d) 48mts
23. If 4x + 3y : 6x + 5y = 11: 17 find x : y
a) 2 : 1 b) 3 : 1 c) 4 : 1 d) 5 : 1 24. If 4p2 + pq : 3pq – q2 = 12 : 5 find p : q
a) 4 : 5 or 3 : 4 b) 3 : 4 or 5 : 7 c) 5 : 6 or 3 : 4 d) None 25. The ages of P and Q are in the ratio 7 : 8, 6 years ago, their ages were in the ratio
5 : 6. Find their present ages a) 21yrs; 28yrs b) 21yrs, 32yrs c) 21yrs; 24yrs d) 28yrs; 32yrs
26. The ratio between two numbers is 5 : 7 and their LCM is 210. Find the numbers
a) 20, 28 b) 30, 42 c) 40, 56 d) 60, 84
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27. In a mixture of 84litres, the ratio of milk and water is 5 : 2. How much water must be added to their mixture 2 : 1? a) 12 litres b) 48litres c) 36litres d) 60litres
28. If p : q = 3 : 5 then 3p2 – 2pq + 5q2: p2 + 7pq – 2q2
a) 61 : 31 b) 61 : 33 c) 61 : 32 d) 61 : 42 29. Divide Rs 2490 among P, Q and R so that 4times P's share; 5times Q's share and
7times R's share may all be equal. What is the value of the difference in share between P and R. a) Rs 550 b) Rs 240 c) Rs 210 d) Rs 450
30. The cost of production of a mobile set is Rs 954. This cost has divided between
Material, Labour and Overheads in the ratio 3 : 4 : 2. Calculate the cost of overhead expenses a) Rs 222 b) Rs 202 c) Rs 212 d) Rs 242
31. What is the ratio whose terms differ by 40 and the measure of which is 27
?
a) 1648
b) 1664
c) 1656
d) 848
32. If 3m – 8 : 2m + 1 is the duplicate ratio of 2 : 3, find the value of m
a) 5 b) 4 c) 6 d) 3 33. If 3c – 9 : 5c + 4 is the triplicate ratio of 3 : 4, find the value of c
a) 14 b) 13 c) 16 d) 18 34. If 2K + 3 : 3k + 7 is the sub duplicate ratio of 9 : 25, find the value of k
a) 8 b) 9 c) 6 d) 7 35. Find the compound ratio of duplicate ratio of 5 : 6, the receprocal of 25 : 32 and
sub duplicate ratio of 81 : 64 a) 1 : 2 b) 2 : 1 c) 1 : 1 d) 3 : 2
36. If 1ab then
a) 1a xb x
b) 1a xb x
c) 1a xb x
d) all of the above
37. If 1ab then
a) a x ab x b
b) a x ab x b
c) a x ab x b
d) (1) and
(2)
38. If a c e kb d f then ......
.....a c eb d f
a) 2k b) 3k c) k d) 1k
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39. If 20% of x is same as 40% of y, then x : y is equal to
a) 1 : 2 b) 1 : 3 c) 3 : 1 d) 2 : 1 40. If x : y = 2 : 3 and 2 : x = 1 : 2 then the value of y is
a) 8 b) 4 c) 6 d) 12 41. Two numbers are in the ratio 5 : 7 if each number is increased by 10 the ratio
becomes 3 : 4. The numbers are a) 40, 56 b) 50, 84 c) 50, 70 d) 60, 84
42. What number must be added to each one of 6, 14, 18, 38 to make it equally
proportionate a) 1 b) 2 c) 3 d) 4
43. The 4th proportional to 0.4, 0.12 and 0.3 is
a) 0.03 b) 0.3 c) 0.9 d) 0.09 44. The third proportional to 0.6 and 0.4 is
a) 154
b) 3.75 c) 415
d) 215
45. The mean proportional between 16 and 36 is
a) 28 b) 24 c) 26 d) 22 46. Two whole numbers whose sum is 64, cannot be in the ratio
a) 5 : 3 b) 7 : 1 c) 3 : 4 d) 9 : 7 47. The ratio of two numbers is 5 : 3 and their sum is 448. The greater than number is
a) 280 b) 270 c) 290 d) 300 48. If 25 : b = 3 : 9 then the value of b is
a) 75 b) 65 c) 85 d) 95 49. If a : b = 18 : 14 then a + b : a – b =
a) 8 : 2 b) 8 : 3 c) 8 : 1 d) 8 : 5
50. If 13
A B and 12
B C then C : A
a) 3 : 6 b) 1 : 6 c) 6 : 3 d) 6 : 1
51. The ratio which ( 15
of Rs 6.30) bears to (0.4 of Rs 1.20) is
a) 21 : 8 b) 8 : 21 c) 21 : 7 d) 21 : 6
52. If 14
th
of A = 16
th
of B = 18
th
of C then C : A : B is
a) 2 : 3 : 4 b) 3 : 2 : 4 c) 4 : 2 : 3 d) 2 : 4 : 3
53. If 1 1 1 1: :4 0.81y y
then the value of y is
a) 0.9 b) 0.27 c) 0.18 d) 1.8
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54. A fractions bears the same ratio to 127
as 37
does to 59
. The fraction is
a) 745
b) 135
c) 457
d) 521
55. If 7 15 17, ,15 20 25
A B C , the smallest ratio is
a) B b) C c) A d) None of these
DIRECT AND INDIRECT PROPORTION
_IMPORTANT FACTS AND FORMULAE 1. Direct Proportion: Two quantities are said to be directly proportional, if on the increase (or decrease) of the one, the other increases (or decreases) to the same Ex. 1. Cost is directly proportional to the number of articles. (More Articles, More Cost) Ex. 2. Work done is directly proportional to the number of men working on it (More Men, More Work)
2. Indirect Proportion: Two quantities are said to be indirectly proportional,if on the increase of the one, the other decreases to the same extent and vice-versa.
Ex. 1. The time taken by a car in covering a certain distance is inversely proportional to the speed of the car.
(More speed, Less is the time taken to cover a distance) Ex. 2. Time taken to finish a work is inversely proportional to the num of persons working at it.
(More persons, Less is the time taken to finish a job) Remark: In solving questions by chain rule, we compare every item with the term to be found out.
TIME AND WORK 1. If A can do a piece of work in n days, then A's 1 day's work = (1/n). 2. If A’s 1 day's work = (1/n),then A can finish the work in n days.
3. A is thrice as good a workman as B, then: Ratio of work done by A and B = 3 : 1. Ratio of times taken by A and B to finish a work = 1 : 3. MCQ QUESTIONS FOR PRACTICE 1. If the cost of 1 kg of sugar is Rs 18, then what would be the cost of 3 kg sugar?
(a) Rs. 54 (b) Rs. 6 (c) Rs. 18 (d) none of these 2. If the cost of 9 toys is Rs. 333, find the cost of 16 such toys.
(a) Rs. 594 (b) Rs. 596 (c) Rs. 592 (d) none of these 3. If 25 metres of cloth costs of Rs. 1575, how many metres of it can be bought for
Rs. 2016?
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(a) 30 m (b) 32 m (c) 36 m (d) none of these 4. A worker is paid Rs. 1110 for 6 days. If his total wages during a month are Rs.
4625, for how many days did he work? (a) 15 days (b) 25 days (c) 30 days (d) none of these
5. A car can cover a distance of 522 km on 36 litres of petrol. How far can it travel on 14 litres of petrol? (a) 230 km (b) 232 km (c) 203 km (d) none of these
6. If 13 metres of a uniform iron rod weighs 23.4 kg then what will be the weight of
6 metres of the same rod? (a) 10 kg (b) 20 kg (c) 10.8 kg (d) none of these
7. The length of the shadow of a 3m high pole at a certain time of the day is 3.6 m.
What is the length of the height of another pole whose shadow at that time is 54 m long? (a) 30 m (b) 40 m (c) 45 m (d) none of these
8. Traveling 900 km by rail costs Rs. 280. What would be the fare for a journey of
360 km when a person travels by the same class? (a) Rs. 118 (b) Rs. 112 (c) Rs. 119 (d) none of these
9. A train covers a distance of 51 km in 45 minutes. How long will it take to cover 221 km?
(a) 3 hours (b) 3 14
hrs (c) 3 12
hrs (d) none of these
10. If 15 oranges cost Rs. 70, what do 39 oranges cost?
(a) Rs. 180 (b) Rs. 182 (c) Rs. 190 (d) none of these
11. If 8 kg sugar costs Rs. 148, how much sugar can be bought for Rs. 832.50? (a) 45 kg (b) 50 kg (c) 60 kg (d) none of these
12. The cost of 37m of silk is Rs. 3145. What length of this silk can be purchased for Rs. 1445? (a) 15 m (b) 16 m (c) 17 m (d) none of these
13. If 22.5 m of a uniform iron rod weighs 85.5 kg, what will be the length of 22.8 kg
of the same rod? (a) 5 m (b) 6 m (c) 7 m (d) none of these
14. If the weight of 6 sheets of a paper is 162 grams, how many sheets of the same
quality of paper would weigh 13.5 kg? (a) 300 (b) 400 (c) 500 (d) none of these
15. 1152 bars of soap can be packed in 8 cartoons of the same size. How many such cartoons will be required to pack 3888 bars? (a) 27 (b) 24 (c) 25 (d) none of these
16. If the thickness of a pile of 16 cardboards is 44mm, how many cardboards will be there in a pile which is 71.5 cm thick?
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(a) 270 (b) 260 (c) 250 (d) none of these
17. At a particular time of a day, a 7m high flagstaff casts a shadow which is 8.2m long. What is the height of the building which casts a shadow 20.5 m in length at the same time? (a) 15.5 m (b) 16.5 m (c) 17.5 m (d) none of these
18. 15 men can build a 16.25m long wall up to a certain height in one day. How many men should be employed to build a wall of the same height but of length 26m in one day? (a) 27 (b) 24 (c) 25 (d) none of these
19. In a hospital, the monthly consumption of milk of 60 patients is 1350 litres. How many patients can be accommodated in the hospital if the monthly ration of milk is raised 1710 litres, assuming that the quota per head remains the same? (a) 75 (b) 76 (c) 77 (d) none of these
20. The extension in an elastic string varies directly as the weigh hung on it. If a weight of 150 g produces an extension of 2.8 cm, what weight would produce an extension of 19.6 cm? (a) 1.5 kg (b) 1.05 kg (c) 15 kg (d) none of these
21. A car travels 432 km on 48 litres. How far would it travel on 20 litres of petrol? (a) 160 km (b) 180 km (c) 200 km (d) none of these
22. If 40m of a cloth costs Rs. 1940, how many metres can be bought for Rs. 727.50?
(a) 15 m (b) 16 m (c) 17 m (d) none of these 23. A private taxi charges a fare of Rs. 260 for a journey of 200 km, how much would
it travel for Rs. 279.50? (a) 200 km (b) 215 km (c) 200 km (d) none of these
24. Manoj types 540 words during half an hour, how many words would he type in 6
minutes? (a) 105 (b) 106 (c) 108 (d) none of these
25. Rohit bought 12 registers for Rs. 156, find the cost of 7 such register.
(a) Rs. 90 (b) Rs. 91 (c) Rs. 92 (d) none of these 26. Pranshu takes 125 minutes in walking a distance of 100m. What distance would
he cover in 315 minutes? (a) 250 m (b) 252 m (c) 254 m (d) none of these
27. If the cost of 93m of a certain kind of plastic sheet is Rs. 1395, then what would it
cost to bury 105m of such plastic sheet? (a) Rs. 1500 (b) Rs. 1550 (c) Rs. 1575 (d) none of these
28. Ranjita types 1080 words in one hour. What is her gross words a minute rate?
(a) 15 (b) 16 (c) 18 (d) none of these 29. 68 boxes of a certain commodity require a shelf-length of 13.6 m. How many
boxes of the same commodity would occupy a shelf-length of 20.4m? (a) 104 (b) 106 (c) 102 (d) none of these
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30. A worker is paid Rs. 200 for 8 days work. If he works for 20 days, how much will
he get? (a) Rs. 500 (b) Rs. 550 (c) Rs. 575 (d) none of these
31. If 52 men can do a piece of work in 35 days, in how many days 28 men will do it?
(a) 65 days (b) 75 days (c) 80 days (d) none of these 32. If 56 men can do a piece of work in 42 days, how many men will do it in 14 days?
(a) 165 (b) 166 (c) 168 (d) none of these 33. 120 men have food provision for 200 days. After 5 days, 30 men died due to an
cancer. How long will the remaining food last? (a) 260 days (b) 275 days (c) 250 days (d) none of these
34. If x and y varies inversely as each other and x = 10 when y = 6. Find y when x =
15. (a) 5 (b) 6 (c) 4 (d) none of these
35. Shreya cycles to her school at an average speed of 12km/hr. It takes her 20
minutes to reach the school. If she wants to reach her school in 15 minutes, what should be her average speed? (a) 15 km/hr (b) 16 km/hr (c) 18 km/hr (d) none of these
36. 1000 soldiers in a fort has enough food for 20 days. But some soldiers were
transferred to another fort and the food lasted for 25 days. How many soldiers were transferred? (a) 100 (b) 200 (c) 800 (d) none of these
37. If x and y varies inversely as each other and x = 8 when y = 32. Find y when x =
16. (a) 64 (b) 16 (c) 4 (d) none of these
38. If x and y varies inversely as each other and x = 8 when y = 10. Find y when x =
2. (a) 40 (b) 16 (c) 4 (d) none of these
39. If x and y varies inversely as each other and x = 2 when y = 40. Find x when x =
20. (a) 40 (b) 16 (c) 4 (d) none of these
40. If a and b varies inversely as each other and a = 16 when b = 4. Find b when a = 8.
(a) 2 (b) 8 (c) 4 (d) none of these 41. If x and y varies directly as each other and x = 4 when y = 16. Find y when x = 9.
(a) 48 (b) 36 (c) 4 (d) none of these 42. If x and y varies directly as each other and x = 4 when y = 16. Find y when x = 12.
(a) 48 (b) 36 (c) 4 (d) none of these 43. If x and y varies directly as each other and x = 4 when y = 16. Find y when x = 1.
(a) 48 (b) 36 (c) 4 (d) none of these
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44. If x and y varies directly as each other and x = 4 when y = 16. Find y when x = 3. (a) 48 (b) 36 (c) 4 (d) none of these
45. If 36 men can do a piece of work in 25 days, in how many days will 15 men do it?
(a) 60 days (b) 75 days (c) 50 days (d) none of these 46. A work force of 50 men with a contractor an finish a piece of work in 5 months.
In how many months the same work can be completed by 125 men? (a) 2 (b) 8 (c) 4 (d) none of these
47. 1200 men can finish a stock of food in 35 days. How many more men should join
the hostel so that the same stock may lst for 25 days? (a) 480 (b) 360 (c) 400 (d) none of these
48. In a hostel of 50 girls, there are food provisions for 40 days. If 30 more girls join
the hostel, how long will these provisions last? (a) 26 days (b) 20 days (c) 25 days (d) none of these
49. 18 men can reap a field in 35 days. For reaping the same field in 15 days, how
many men are required? (a) 48 (b) 36 (c) 42 (d) none of these
50. 55 cows can graze a field in 16 days. How many cows will graze the same field in
10 days? (a) 88 (b) 66 (c) 44 (d) none of these
TIME AND DISTANCE IMPORTANT FACTS AND FORMULAE
Distance ;
Distance ;
Distance
SpeedTime
TimeSpeed
Speed Time
1. 5/ /
18x km hr x m s
2. 18/ /5
x m s x km hr
4. If the ratio of the speeds of A and B is a:b , then the ratio of the times taken by
them to cover the same distance is 1 1: :or b aa b
5. Suppose a man covers a certain distance at x km/ hr and an equal distance at y km / hr . Then , the average speed during the whole journey is 2 /xy km hr
x y
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MCQ QUESTIONS FOR PRACTICE 1. If a person walks at 14 km/hr instead of 10 km/hr, he would have walked 20 km
more. The actual distance travelled by him is:? a)50 km b)56 km c)70 km d)80 km
2. The speed of a car increases by 2 kms after every one hour. If the distance
travelling in the first one hour was 35 kms. what was the total distance travelled in 12 hours?? a)456 kms b)482 kms c)552 kms d)556 kms
3. A person travels from P to Q at a speed of 40 kmph and returns by increasing his
speed by 50%. What is his average speed for the both the trips?? a)36 kmph b)40 kmph c)45 kmph d)48 kmph
4. Two trains 142 meter & 138 meter is length are running towards each other on
parallel line one at rate of 32 km/hr and another at the rate of 40 km/hr in what time they will clear off each other.? a)13 b)14 c)15 d)16
5. A train covers a distance in 50 min, if it runs at a speed of 48kmph on an average.
The speed at which the train must run to reduce the time of journey to 40min will be.? a)45 kmph b)60 kmph c)75 kmph d)80 kmph
6. A train can travel 50% faster than a car. Both start from point A at the same time
and reach point B 75 kms away from A at the same time. On the way, however, the train lost about 12.5 minutes while stopping at the stations. The speed of the car is:? a)100 kmph b)110 kmph c)120 kmph d)130 kmph
7. A person croses a 600 m long street in 5 minutes. What is his speed in km per
hour?? a)3.6 b)7.2 c)8.4 d)10
8. A person has to cover a distance of 6 km in 45 minutes. If he covers one-half of
the distance in two-thirds of the total time, to cover the remaining distance in the remaining time, his speed (in km/hr) must be:? a)6 b)8 c)12 d)15
9. A person during his journey travels it in three parts, with speed of 10km/hr, 60
km/hr and 30 km/hr respectively. Find his average speed for the whole journey? a)33.33 b)50 c)20 d)32.5
10. Sachin can cover a distance in 1hr 24min by covering 2/3 of the distance at 4
kmph and the rest at 5kmph.the total distance is?? a)5 km b)6 km c)7 km d)8 km
11. It takes eight hours for a 600 km journey, if 120 km is done by train and the rest
by car. It takes 20 minutes more, if 200 km is done by train and the rest by car. The ratio of the speed of the train to that of the cars is: ? a)2 : 3 b)3 : 2 c)3 : 4 d)4 : 3
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12. A car moves at the speed of 80 km/hr. what is the speed of the car in metres per second? ?
a)8 m/sec b)20 19
m/sec c)21 29
m/sec d)22 29
m/sec
13. A train starts from Delhi at 6:00 am and reaches Ambala cantt. at 10am. The other
train starts from Ambala cantt. at 8am and reached Delhi at 11:30 am, If the distance between Delhi and Ambala cantt is 200 km, then at what time did the two trains meet each other? a)8:46 am b)8:30 am c)8:56 am d)8:50 am
14. A farmer travelled a distance of 61 km in 9 hours. He travelled partly on foot @ 4 km/hr and partly on bicycle @ 9 km/hr. The distance travelled on foot is: ? a)14 km b)15 km c)16 km d)17 km
15. A and B walk a circulat track. They start at 8 a.m from the same point in the opposite directions. A and B walk at a speed 0f 2 rounds per hour and 3 rounds per hour respectively. How many times shall they cross each other before 9.30 a.m? ? a)5 b)7 c)9 d)11
16. Two stations A and B are 110 km apart on a straight line. One train starts from A
at 7 am and travel towards B at 20 km/hr speed. Another train starts from B at 8 am and travel towards A at 25 km/hr speed. At what time will they meet?? a)9 am b)10 am c)11 am d)None of these
17. A man covered a certain distance at some speed. Had he moved 3 kmph faster, he
would have taken 40 minutes less. If he had moved 2 kmph slower, he would have taken 40 minutes more. The distance (in km) is:? a)35 b)37 c)39 d)40
18. The distance between two cities A and B is 330 Km. A train starts from A at 8 a.m. and travel towards B at 60 km/hr. Another train starts from B at 9 a.m and travels towards A at 75 Km/hr. At what time do they meet? ? a)10 a.m b)10.30 a.m c)11 a.m d)11.30 a.m
19. A man can row 4.5 km/hr in still water and he finds that it takes him twice as long to row up as to row down the river. Find the rate of the stream. ? a)2 km/hr b)2.5 km/hr c)1.5 km/hr d)1.75 km/hr
20. A man in a train notices that he can count 21 telephone posts in one minute. If they are known to be 50 metres apart, then at what speed is the train travelling? ? a)55 km/hr b)57 km/hr c)60 km/hr d)63 km/hr
21. Mac travels from A to B a distance of 250miles in 5½ hours. He returns to A in 4
hours 30 minutes. His average speed is: (a) 42 mph (b) 49 mph (c) 48 mph (d) 50 mph
22. A boy goes to his school from his house at a speed of 3 km/hr and returns at a
speed of 2 km/hr. If he takes 5 hours in going and coming, the distance between
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his house and school is: (a) 8.5 km (b) 5.5 km (c) 6 km (d) 9 km
23. The average speed of a train in the onward journey is 25% more than that in the
return journey. The train halts for one hour on reaching the destination. The total time taken for the complete to and for journey is 17 hours, covering a distance of 800 km. The speed of the train in the onward journey is: (a) 50 km/hr (b) 53 km/hr (c) 52 km/hr (d) 56.25 km/hr
24. A man on tour travels first 160 km at 64 km/hr and the next 160 kin at 80 km/hr.
The average speed for the first 320 km of the tour is: (a) 35.55 km/hr (b) 38 km/hr (c) 71.11 km/hr (d) 75 km/hr
25. A boy rides his bicycle 10 km at an average speed of 12 km/hr and again travels
12 km at an average speed of 10 km/hr. His average speed for the entire trip is approximately : (a) 10.4 km/hr (b) 10.8 km/hr (c) 12 km/hr (d) 14 km/hr
26. A car travels the first one-third of a certain distance with a speed of 10 km/hr, the
next one- third distance with a speed of 20 km/hr, and the last one-third distance with a speed. of 60 km/hr. The average speed of the car for the whole journey is: (a) 18 km/hr (b) 34 km/hr (c) 35 km/hr (d) 39 km/hr
27. A motorist covers a distance of 39 km in 45 minutes by moving at a speed of x
kmph for the first 15minutes, thenmoving at double the speed for the next 20 minutes and then again moving at his original speed for the rest of the journey Then, x is equal to: (a) 31.2 (b) 36 (c) 42 (d) 54
28. Mary jogs 9 km at a speed of 6 km per hour. At what speed would she need to jog
during the next 1.5 hours to have an average of 9 km per hour for the entire jogging session? (a) 9 kmph (b) 13 kmph (c) 12 kmph (d) 15 kmph
29. A car traveling with 5/7 of its actual speed covers 42 km in 1 hr 40min 48 sec.
Find the actual speed of the car. (a) 17 km/hr (b) 32 km/hr (c) 31 km/hr (d) 35 km/hr
30. A train running at 7/11 of its own speed reached a place in 22 hours. How much
time could be saved if the train would have run at its own speed? (a) 9 hours (b) 8 hours (c) 14 hours (d) 20 hours
31. A man can reach a certain place in 30 hours. If he reduces his speed by 1/15th, he
goes 10 km less in that time. Find his speed. (a) 7 km/hr (b) 5 km/hr (c) 52 km/hr (d) 8 km/hr
32. Walking 6/7th of his usual speed, a man is 12 minutes too late. The usual time
taken by him to cover that distance is: (a) 2 hour (b) 1 hr 12 min. (c) 1 hr 15 min. (d) 3 hr 20 min
33. A train when moves at an average speed of 40 kmph, reaches its destination on
time. When its average speed becomes 35 kmph, then it reaches its destination 15
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minutes late. Find the length of journey. (a) 50 km (b) 60 km (c) 70 km (d) 90 km
34. Robert is traveling on his cycle and has calculated to reach point A at 2 PM. if he
travels at 10 kmph; he will reach there at 12 noon if he travels at 15 kmph. At what speed must he travel to reach A at 1 P.M.? (a) 8 kmph (b) 13 kmph (c) 12 kmph (d) 16 kmph
35. If a train runs at 40 kmph, it reaches its destination late by 11minutes but if it runs
at 50 kmph, it is late by 5 minutes only. The correct time for the train to complete its journey is: (a) 13 min. (b) 17 min. (c) 19 min. (d) 22 min
36. A man covered a certain distance at some speed. Had he moved 3 kmph faster, he
would have taken 40 minutes less. If he had moved 2 kmph, slower, he would have taken 40 minutes more. The distance (in km) is : (a) 353 (b) 368 (c) 372 (d) 40
37. What is the time taken by a train running at 18km/h to cross a man standing on a
platform, the length of the train being 120 m? (a) 22 s (b) 44 s (c) 12 s (d) 24 s
38. A man is walking at a speed of 10 km/h. After every kilometer, he takes rest for
5min.Howmuch time will he take to cover a distance of 5 km? (a) 20 min (b) 30 min (c) 40 min (d) 50 min
39. Walking 5/7 of his usual rate, a boy reaches his school 6 min late. Find his usual
time to reach school. (a) 10 min (b) 12 min (c) 15 min (d) 18 min
40. If I walk at 4 km/h, I miss the bus by 10 min. If I walk at 5 km/h, I reach 5 min
before the arrival of the bus. How far I walk to reach the busstand? (a) 5 km (b) 5.5 km (c) 6 km (d) 7.5 km
SIMPLE AND COMPOUND INTEREST
IMPORTANT FACTS AND FORMULAE
1.. Principal: The money borrowed or lent out for a certain period is called the principal or the sum.
2. Interest: Extra money paid for using other's money is called interest. 3. Simple Interest (S.I.) : If the interest on a sum borrowed for a certain period is reckoned uniformly, then it is called simple interest. Let Principal = P, Rate = R% per annum (p.a.) and Time = T years. Then,
(i) . .100
P R TS I
(ii) 100 100 100; ;I I IP R TR T P T P R
Compound Interest: Sometimes it so happens that the borrower and the lender agree to fix up a certain unit of time, say yearly or half-yearly or quarterly to settle the
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previous account.
In such cases, the amount after first unit of time becomes the principal for the second unit,the amount after second unit becomes the principal for the third unit and so on.
After a specified period, the difference between the amount and the money borrowed is called the Compound Interest (abbreviated as C.I.) for that period.
Let Principal = P, Rate = R% per annum, Time = n years. I. When interest is compound Annually:
Amount = 1100
nRP
II. When interest is compounded Half-yearly:
Amount = 2
1200
nRP
III. When interest is compounded Quarterly:
Amount = 4
1400
nRP
IV. When interest is compounded Annually but time is in fraction, say
235
years.
Amount = 3 21 1
100 500R RP
V. When Rates are different for different years, say Rl%, R2%, R3% for 1st, 2nd and 3rd year respectively.
Then, Amount = 31 21 1 1100 100 100
RR RP
VI. Present worth of Rs.x due n years hence is given by :
Present Worth = 1
100
nxR
MCQ QUESTIONS FOR PRACTICE 1. The simple interest on Rs. 5000 for 219 days at 4% per annum is
a) Rs 126.50 b) Rs 120 c) Rs 125 d) Rs. 43.80 2. Find the simple interest on Rs. 2500 for 2 years 6 months at 6% per annum.
a) Rs 350 b) Rs 375 c) Rs 750 d) none of these 3. What sum will amount to Rs. 4590 at 12% per annum simple interest in 3 years?
a) Rs 3500 b) Rs 3375 c) Rs 3750 d) none of these
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4. In what time will Rs. 1860 amount to Rs. 2278.50, if simple interest is calculated
at 9% per annum?
a) 2 years b) 2 12
years c) 3 years d) 4 years
5. At what rate percent per annum will Rs. 1650 amount to Rs. 2046 in 3 years?
a) 8% b) 4% c) 6% d) 16 23
%
6. Simple interest on a certain sum is 1625
of the sum. Find the rate percent and the
time if both are numerically equal.
a) 6 years b) 2 12
years c) 8 years d) 4 years
7. At what rate percent per annum simple interest will a sum treble itself in 16 years?
a) 12% b) 12.5% c) 15% d) 16 23
%
8. A sum of money at simple interest amounts to Rs. 696 in 2 years and Rs. 840 in 5
years. The sum is a) Rs 500 b) Rs 600 c) Rs 560 d) Rs. 620
9. In what time will Rs. 1600 amount to Rs. 1768 at 6% per annum simple interest?
a) 1 14
years b) 2 12
years c) 1 34
years d) 1 12
years
10. A sum amounts to Rs. 3720 in 8 months at 5% per annum simple interest. The
sum is a) Rs 3500 b) Rs 3600 c) Rs 3560 d) Rs. 3620
11. At what rate percent per annum simple interest will a sum be double itself in 8
years?
a) 15% b) 14% c) 16% d) 12 12
%
12. At simple interest a sum becomes of itself in 5 years. The rate of interest percent
per annum is a) 8% b) 5% c) 10% d) 12%
13. What amount Heena has to pay at the end of 2 years on a sum of Rs 20,000 at an
interest of 8% compounded annually? a) Rs 23328 b) Rs 23000 c) Rs 23350 d) Rs
23450 14. What will be the bill amount if the cost of a toy is Rs 450 and the sales tax charge
is 5%? a) Rs 472.5 b) Rs 437.5 c) Rs 460 d) Rs 455
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15. The compounded interest on Rs. 50,000 at 8% per annum for 2 years,
compounded annually is a) Rs 8000 b) Rs 8250 c) Rs 8350 d) Rs 8640
16. At what rate percent per annum will a sum of Rs. 7500 amount to Rs. 8427 in 2
years compounded annually? a) 5% b) 4% c) 6% d) 8%
17. The compounded interest on Rs. 30000 at 7% per annum for a certain time is Rs.
4347. The time is
a) 2 years b) 2 12
years c) 3 years d) 4 years
18. The principal that amounts to Rs. 4913 in 3 years at 6 14
% per annum compound
interest, compounded annually is a) Rs 3096 b) Rs 4076 c) Rs 4085 d) Rs 4096
19. If the simple interest on a sum of money at 5% per annum for 3 years is Rs. 1200,
then the compound interest on the same sum for he same period at the same rate will be a) Rs 1225 b) Rs 1236 c) Rs 1248 d) Rs 1261
20. If the compound interest on a sum for 2 years at 12 12
% per annum is Rs. 510, the
simple interest on the same sum at the same rate for same period of time is a) Rs 400 b) Rs 450 c) Rs 460 d) Rs 480
21. The difference between compound interest and simple interest on a sum of Rs.
15000 for 2 years is Rs. 96. What is the rate of interest percent per annum? a) 10% b) 12% c) 6% d) 8%
22. The compounded interest on Rs. 8,000 at 5% per annum for 3 years, compounded
annually is a) Rs 1225 b) Rs 1236 c) Rs 1248 d) Rs 1261
23. The compounded interest on Rs. 6400 at 7 12
% per annum for 2 years,
compounded annually is a) Rs 7096 b) Rs 7396 c) Rs 7000 d) none of these
24. In what time will be Rs. 1000 amount to Rs. 1331 at 10% per annum compounded
annually?
a) 2 years b) 2 12
years c) 3 years d) 4 years
25. The yearly interest payable on a deposit of Rs. 250 at 5.5% p.a. simple interest is: a) Rs. 137.50 b) Rs. 13.75 c)Rs. 12.50 d)Rs. 125.00 e)Rs. 17.35
26. The interest on Rs. 12 167 invested for 5 years at 2.5% simple interest p.a. would
be nearest to: a)Rs. 1220.87 b)Rs. 1521 c)Rs. 1520.88 d)Rs. 152.10 e)Rs. 152.09
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27. Sarah received Rs. 75 interest on her term deposit of Rs. 450 over 3 years. The simple interest rate per annum was: a)56% b)5.6% c)22% d)0.55% e)11.2%
28. Tom earned Rs. 504 when he invested Rs. 16 820 for 8 months. His rate of simple
interest was: a)0.37% p.a. b)2.00% p.a. c)3.00% p.a. d)4.49% p.a. e)5.00% p.a.
29. John invested a sum of money, which earned Rs. 750 simple interest at the rate of
6% per annum over 2 years. The amount invested was: a)Rs. 88.80 b)Rs. 6250 c)Rs. 3125 d)Rs. 2125 e)None of these.
30. Cathy invested Rs. 8200 at the rate of 4.5% p.a. It earned Rs. 738 simple interest. The period of investment was: a)6 months b)1 year c)2 years d)2 years 6 months e)3 years
31. A 500-g packet of chocolate costs Rs. 2.50. Assuming inflation averages 2.8% per
annum over each of the next 3 years, how much will the chocolate cost in three years? a)Rs. 2.64 b)Rs. 2.72 c)Rs. 2.79 d)Rs. 2.87 e)Rs. 2.95
32. An investment of Rs. 10 000 at the rate of 8% per annum, compounded quarterly,
will reach Rs. 14 800 in close to: a)1 year b)2 years c)3 years d)4 years e)5 years
33. An investment of Rs. 10 000 to be invested for a period of 5 years and
compounded quarterly at 3.75% p.a. will have a future value closest to which of the following: a)Rs. 10 480 b)Rs. 12 020 c)Rs. 12 050 d)Rs. 20 900 e)None of the
above 34. Tom has the choice of investing his money in compound interest with rests as
indicated below. Which situation would return him the most interest? a)yearly rests b)six-monthly rests c)quarterly rests d)monthly rests e)daily rests
35. The growth of an investment over a period of time is shown as a graph which
depicts a straight line. Which of the following investments would depict a straight line? a)6% p.a. simple interest b)6% p.a. compounding yearly c)6% p.a. compounding quarterly d)6% p.a. compounding monthly e)6% p.a. compounding daily
36. How long would it take for Rs. 5000 invested at 5% p.a. compound interest with
yearly rests to double in value? a)5 years b)7 years c)10 years d)14 years e)20 years
37. An interest rate of 4.5% p.a. compounding monthly is equivalent to an effective
interest rate of: a)4.50% p.a. b)4.55% p.a. c)4.57% p.a. d)4.59% p.a. e)4.60% p.a.
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38. Which of the following compounding rates is equivalent to an effective interest rate of 2.75% p.a.? a)2.7% p.a. compounding six-monthly b)2.75% p.a. compounding yearly c)2.6% p.a. compounding quarterly d)2.6% p.a. compounding monthly e)2.6% p.a. compounding daily
39. Two banks pay simple interest on short-term deposits. Bank A pays 5% p.a. over
3 years, and Bank B pays 5.5% p.a. for 2 21 years. The difference between the two
banks’ final payout figure, if Rs. 10 000 was invested in each account is: a)Rs. 0 b)Rs. 125 c)Rs. 250 d)Rs. 1375 e)Rs. 1500
40. Paul wishes to invest Rs. 10 000 for a period of 5 years. Which of the following
investments would be best for him? a)6.7% p.a. simple interest b)6.75% p.a. compound interest with yearly rests c)6.5% p.a. compound interest with quarterly rests d)6.25% p.a. compound interest with monthly rests e)6% compound interest with daily rests.
41. An investment of Rs. 6000 was placed for 3 years at 4.25% p.a. compounded
annually. How much more would be collected if the investment was compounded quarterly? a)No more b)Rs. 13.36 c)Rs. 32.97 d)Rs. 46.33 e)Rs. 52.29
42. An automobile financier claims to be lending money at simple interest, but he
includes the interest every six months for calculating the principal. If he is charging an interest of 10%, the effective rate of interest becomes: a)13% b)10.25% c)15% d)11% e)None of these
43. A sum of money at simple interest amounts to Rs. 815 in 3 years and to Rs. 854 in 4 years. The sum is: a)Rs. 850 b)Rs. 790 c)Rs. 698 d)Rs. 800 e)Rs. 600
44. Sum of money becomes Rs. 13,380 after 3 years and Rs. 20,070 after 6 years on compound interest. The sum is: a)Rs. 9200 b)Rs. 9000 c)Rs. 8920 d)Rs. 9040 e)Rs. 9500
45. A sum of Rs. 12,000 deposited at compound interest becomes double after 5 years. After 20 years, it will become: a)Rs. 1,10,000 b)Rs. 1,30,000 c)Rs. 1,24,000 d)Rs. 1,92,000 e)Rs. 1,50,000
46. A sum of money placed at compound interest doubles itself in 5 years. It will
amount to eight times itself at the same rate of interest in: a)7 years b)12 years c)15 years d)30 years e)21 years
47. If a sum on compound interest becomes three times in 4 years, then with the same
interest rate, the sum will become 27 times in: a)11 years b)12 years c)24 years d)38 years e)21 years
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48. The least number of complete years in which a sum of money put out at 20% compound interest will be more than doubled is: a)7 b)4 c) 5 d) 8 e) 7
49. A man borrows Rs. 2550 to be paid back with compound interest at the rate of
4% per annum by the end of 2 years in two equal yearly installments. How much will each installment be? a)Rs.1275 b)Rs.1383 c) Rs.1352 d) Rs.1287 e) Rs.1250
50. What annual payment will discharge a debt of Rs. 1025 due in 2 years at the rate of 5% compound interest? a)Rs.650 b) Rs.551.25 c) Rs.560 d) Rs.660.75 e) Rs.600
51. The difference between compound interest and simple interest on an amount of
Rs. 15,000 for 2 years is Rs. 96. What is the rate of interest per annum? (a) 8 (b) 11 (c) 12 (d) None of these
52. The difference between simple and compound interests and compounded annually
on a certain sum of money for 2 years at 4% per annum is Re. 1. The sum (in Rs.) is: (a) 625 (b) 620 (c) 640 (d) 660
53. The compound interest on a sum of money for 2 years is Rs. 832 and the simple
interest on the same sum for the same period is Rs. 800. The difference between the compound interest and the simple interest for 3 years will be: (a) Rs. 50 (b) Rs. 67 (c) Rs. 98.56 (d) Rs. 75.45
54. The difference between the simple interest on a certain sum at the rate of 10% per
annum for 2 years and compound interest which is compounded every 6 months is Rs. 124.05. What is the principal sum? (a) Rs. 9000 (b) Rs. 8000 (c) Rs. 10,000 (d) Rs. 13,000
55. The difference between the simple interest on a certain sum at the rate of 10% per
annum for 2 years and compound interest which is compounded every 6 months is Rs. 124.05. What is the principal sum? (a) Rs. 9000 (b) Rs. 8000 (c) Rs. 10,000 (d) Rs. 13,000
56. A sum of money lent at compound interest for 2 years at 20% per annum would
fetch Rs. 482 more, if the interest was payable half-yearly than if it was payable annually. The sum is: (a) Rs. 12,000 (b) Rs. 20,000 (c) Rs. 40,000 (d) Rs. 60,000
57. On a sum of money, the simple interest for 2 years is Rs. 660, while the
compound interest is Rs. 696.30, the rate of interest being the same in both the cases. The rate of interest is: (a) 13% (b) 14% (c) 12% (d) 11%
58. Sum of money becomes Rs. 13,380 after 3 years and Rs. 20,070 after 6 years on
compound interest. The sumis: (a) Rs. 9200 (b) Rs. 9000 (c) Rs. 8920 (d) Rs. 9040
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59. A sum of Rs. 12,000 deposited at compound interest becomes double after 5 years. After 20 years, it will become: (a) Rs. 1,10,000 (b) Rs. 1,30,000 (c) Rs. 1,24,000 (d) Rs. 1,92,000
60. A sum of money placed at compound interest doubles itself in 5 years. It will
amount to eight times itself at the same rate of interest in: (a) 7 years (b) 12 years (c) 15 years (d) 30 years
MENSURATION
IMPORTANT CONCEPTS OF AREA
I.RESULTS ON TRIANGLES: 1.Sum of the angles of a triangle is 180 degrees. 2.Sum of any two sides of a triangle is greater than the third side. 3.Pythagoras theorem: In a right angle triangle, (Hypotenuse)2 = (base)2 + (Height)2 4.The line joining the midpoint of a side of a triangle to the opposite vertex is called the MEDIAN 5.The point where the three medians of a triangle meet is called CENTROID. Centroid divides each of the medians in the ratio 2:1. 6.In an isosceles triangle, the altitude from the vertex bi-sects the base 7.The median of a triangle divides it into two triangles of the same area. 8.Area of a triangle formed by joining the midpoints of the sides of a given triangle is one-fourth of the area of the given triangle. II.RESULTS ON QUADRILATERALS:
1. The diagonals of a parallelogram bisects each other . 2. Each diagonal of a parallelogram divides it into two triangles of the same area 3. The diagonals of a rectangle are equal and bisect each other. 4. The diagonals of a square are equal and bisect each other at right angles. 5. The diagonals of a rhombus are unequal and bisect each other at right angles. 6. A parallelogram and a rectangle on the same base and between the same
parallels are equal in area. 7. Of all the parallelograms of a given sides, the parallelogram which is a
rectangle has the greatest area.
IMPORTANT FORMULAE
I.1.Area of a rectangle=(length x breadth) Therefore length = (area/breadth) and breadth=(area/length) 2.Perimeter of a rectangle = 2 x (length + breadth)
II.Area of a square = (side)2 = 12
(diagonal)2
III Area of four walls of a room = 2x(length + breadth)x(height)
IV 1.Area of the triangle= 12
(base x height)
2. Area of a triangle = ( )( )( )s s a s b s c , where a, b, c are the sides of a
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triangle and s = 2
a b c
3.Area of the equilateral triangle = 23 ( )4
side
4.Radius of incircle of an equilateral triangle of side a = 32a
5.Radius of circumcircle of an equilateral triangle of side a = 3
a
V.1.Area of the parellogram =(base x height)
2.Area of the rhombus = 12
(product of the diagonals)
3.Area of the trapezium = 12
(size of parallel sides) x distance between them
VI 1.Area of a circle = 2r ,where r is the radius 2. Circumference of a circle = 2 r .
3. Length of an arc = 2360
r where θ is the central angle
4. Area of a sector = 2
360r = 1 ( )
2arc r
VII. 1. Area of a semi-circle = 2
2r
2. Circumference of a semi-circle = r . IMPORTANT FORMULAE ON VOLUME AND SURFACE AREAS I. CUBOID
Let length = 1, breadth = b and height = h units. Then, 1. Volume = (1 x b x h) cubic units. 2. Surface area= 2(lb + bh + lh) sq.units. 3. Diagonal.= 2 2 2l b h units II. CUBE
Let each edge of a cube be of length a. Then, 1. Volume = a3 cubic units. 2. Surface area = 6a2 sq. units. 3. Diagonal = 3a units. III. CYLINDER
Let radius of base = r and Height (or length) = h. Then, 1. Volume = 2r h cubic units. 2. Curved surface area = 2 rh units. 3. Total surface area = 2 ( )r r h sq. units IV. CONE
Let radius of base = r and Height = h. Then, 1. Slant height, 2 2l r h
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2. Volume = 213
r h cubic units.
3. Curved surface area = rl sq. units. 4. Total surface area = ( )r r l sq. units. V. SPHERE
Let the radius of the sphere be r. Then,
1. Volume = 343
r cubic units.
2. Surface area = 24 r sq. units.
VI. HEMISPHERE Let the radius of a hemisphere be r. Then,
1. Volume = 323
r cubic units.
2. Curved surface area = 22 r sq. units. 3. Total surface area = 23 r units. Remember: 1 litre = 1000 cm3 and 1m3 = 1000 litres 1. The area of a rectangular sheet is 500 cm2. If the length of the sheet is 25 cm, what
is its width? (a) 20 cm (b) 17 cm (c) 30 cm (d) 25 cm 2. If the area of rectangle increases from 2 cm2 to 4 cm2 the perimeter will
(a) increase (b) decrease (c) remains same (d) none of these 3. Which figure encloses more area : a square of side 2 cm ; a rectangle of side 3
cm & 2 cm ;An equilateral triangle of side 4 cm (a) rectangle (b) square (c) triangle (d) same of rectangle & square
4. ABC is isosceles in which AE BC, AE = 6 cm , BC = 9 cm, the area of
ABC is (a) 27 cm2 (b) 54 cm2 (c) 22.5 cm2 (d) 45 cm2
5. Which of the following has the formula : Base x Height (a) area of parallelogram (b) area of quadrilateral
(c) area of triangle (d) area of trapezium 6. If the area of the triangle is 36 cm2 and the height is 3 cm, the base of the triangle
will be (a) 12 cm (b) 39 cm (c) 108 cm (d) 24 cm
7. The height of parallelogram whose area is 35 cm2 and altitude 7 cm
(a) 5 cm (b) 5 cm2 (c) 245 cm (d) 245 cm2
8. Area of triangle whose base is 15 cm and corresponding altitude is 6 cm will be
(a) 45 cm2 (b) 90 cm2 (c) 45 cm (d) 90 cm 9. Find the area of a right triangle whose base is 3 cm, perpendicular is 2 cm and
hypotenuse is 5 cm. (a) 3 cm2 (b) 7.5 cm2 (c) 5 cm2 (d) 6 cm
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10. What will be the area of circular button of radius 7 cm
(a) 154 cm2 (b) 49 cm2 (c) 154 cm (d) 3.14 x 7 cm2
11. The circumference of circle whose diameter is 14 cm will be
(a) 44 cm (b) 88 cm (c) 44 cm2 (d) 88 cm2
12. If the area of circle is 44 cm2 , the area of shaded portion will be (a) 11 cm2 (b) 11 cm
(c) 22 cm2 (d) 22 cm2
13. If the radius of pipe is 1 cm , the circumference of pipe will be
(a) 62.8 cm (b) 6.28 cm (c) 62.8 cm2 (d) 6.28 cm
14. One hectare is equal to
(a) 100 m2 (b) 1000 m2 (c) 10,000 m2 (d) 10,000 m 15. The circumference of a circle with radius 7 cm is
(a) 11 cm (b) 22 cm (c) 44 cm (d) 49 cm 16. The area of a circle is 49 cm2. Its circumference is
(a) 7 cm (b) 14 cm (c) 21 cm (d) 28 cm 17. The perimeter of circular field is 242cm. The area of the field is
(a) 9317 cm2 (b) 18634 cm2 (c) 4658.5 cm2 (d) none of these 18. The area of a circle is 38.5 cm2. Its circumference is
(a) 62 cm (b) 12.1 cm (c) 11 cm (d) 22 cm 19. The difference between the circumference and radius of a circle is 37 cm. The area
of the circle is (a) 111 cm2 (b) 184 cm2 (c) 154 cm2 (d) 259 cm2
20. The circumference of two circles are in the ratio 2 : 3. The ratio of their areas is
(a) 2 : 3 (b) 4 : 9 (c) 9 : 4 (d) none of these 21. The area of the square is the same as the area of the circle. Their perimeter re in
the ratio (a) 1 : 1 (b) : 2 (c) 2 : (d) none of these
22. The areas of the two circle are in the ratio 4 : 9. The ratio of their circumference is
(a) 2 : 3 (b) 4 : 9 (c) 9 : 4 (d) 4 : 9 23. In making 1000 revolutions, a wheel covers 88 km. The diameter of the wheel is
(a) 14 m (b) 24 m (c) 28 m (d) 40 m 24. The diameter of a wheel is 40 cm. How many revolutions will it make an covering
176 m? (a) 140 (b) 150 (c) 160 (d) 166
25. The radius of wheel is 0.25 m. How many revolutions will it make in covering 11
km?
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(a) 2800 (b) 4000 (c) 5500 (d) 7000 26. Find the area of a circle whose circumference is 52.8 cm.
(a) 221.76 cm2 (b) 220.76 cm2 (c) 200.76 cm2 (d) none of these. 27. A steel wire when bent in the form of a square, encloses an area of 121 sq. cm.
The same wire is bent in the form of a circle. Find the area of the circle. (a) 111 cm2 (b) 184 cm2 (c) 154 cm2 (d) 259 cm2
28. If the perimeter of a semicircular protractor is 36 cm, find the diameter.
(a) 14 cm (b) 16 cm (c) 18 cm (d) 12 cm 29. A bicycle wheel makes 5000 revolutions in moving 11 km. Find the diameter of
the wheel. (a) 60 cm (b) 70 cm (c) 66 cm (d) 68 cm
30. The diameter of the wheels of a bus is 140 cm. How many revolutions per minute
must a wheel make in order to move a t a speed of 66km/hr? (a) 240 (b) 250 (c) 260 (d) 270
31. The perimeter of regular polygon is (a) no. of sides x lengths of one side (b) no. of sides + lengths of one side
(c) no. of sides – lengths of one side (d) no. of sides lengths of one side 32. A wire is in the shape of a square of side 10 cm. If the wire is rebent into a
rectangle of length 12 cm, find its breadth. (a) 12 cm (b) 7 cm (c) 8 cm (d) 9 cm
33. A paper is in the form of a rectangle ABCD in which AB = 18cm and BC = 14cm.
A semicircular portion with BC as diameter is cut off. Find the area of the remaining paper (see in below figure). (a) 175 cm2 (b) 165 cm2 (c) 145 cm2 (d) none of these
34. Find the area of the shaded region in the above sided figure. Take = 3.14
(a) 75 cm2 (b) 72 cm2 (c) 70 cm2 (d) none of these 35. The area of a square and a rectangle are equal. If the side of the square is 40 cm
and the breadth of the rectangle is 25 cm, find the length of the rectangle. (a) 60 cm (b) 62 cm (c) 64 cm (d) 68 cm
36. The perimeter of floor of rectangular hall is 250m. The cost of the white washing
its four walls is Rs. 15000. The height of the room is (a) 5m (b) 4m (c) 6m (d) 8m
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37. The perimeter of parallelogram PQRS is: (a) 12 cm (b) 7 cm (c) 38 cm (d) 19 cm
P Q
S R
7 cm
12 cm
38. The length, breadth and height of a room is 5m, 4m and 3m. The cost of white
washing its four walls at the rate of Rs. 7.50 per m2 is (a) Rs. 110 (b) Rs. 109 (c) Rs. 220 (d) Rs. 105
39. The breadth of a room is twice its height and is half of its length. The volume of
room is 512dm3. Its dimensions are (a) 16 dm, 8 dm, 4 dm (b) 12 dm, 8 dm, 2 dm (c) 8 dm, 4 dm, 2 dm (d) 10 dm, 15 dm, 20 dm
40. The area of three adjacent faces of a cube is x, y and z. Its volume V is
(a) V = xyz (b) V3 = xyz (c) V2 = xyz (d) none of these
41. Two cubes each of edge 12 cm are joined. The surface area of new cuboid is (a) 140 cm2 (b) 1440 cm2 (c) 144 cm2 (d) 72 cm2
42. The curved surface area of a cylinder of height 14 cm is 88 cm2. The diameter of
its circular base is (a) 5cm (b) 4cm (c) 3cm (d) 2cm
43. It is required to make a closed cylindrical tank of height 1 m and base diameter
140cm from a metal sheet. How many square meters a sheet are required for the same? (a) 6.45m2 (b) 6.48m2 (c) 7.48m2 (d) 5.48m2.
44. A metal pipe is 77 cm long. Inner diameter of cross section is 4 cm and outer
diameter is 4.4 cm. Its inner curved surface area is: (a) 864 cm2 (b) 968 cm2 (c) 768 cm2 (d) none of these
45. The diameter of a roller is 84 cm and its length is 120 cm. It takes 500 complete
revolutions to move once over to level a playground. The area of the playground in m2 is: (a) 1584 (b) 1284 (c) 1384 (d) 1184
46. A cylindrical pillar is 50 cm in diameter and 3.5 m in height. The cost of painting
its curved surface at the rate of Rs. 12.50 per m2 is: (a) Rs. 68.75 (b) Rs. 58.75 (c) Rs. 48.75 (d) Rs. 38.75
47. The inner diameter of circular well is 3.5m. It is 10m deep. Its inner curved
surface area in m2 is: (a) 120 (b) 110 (c) 130 (d) 140
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48. In a hot water heating system there is a cylindrical pipe of length 28 m and diameter 5 cm. The total radiating surface area in the system in m2 is: (a) 6.6 (b) 5.5 (c) 4.4 (d) 3.4
49. The curved surface area of a right circular cone of slant height 10 cm and base
radius 7 cm is (a) 120 cm2 (b) 220 cm2 (c) 240 cm2 (d) 140 cm2
50. The height of a cone is 16 cm and base radius is 12 cm. Its slant height is
(a) 10 cm (b) 15 cm (c) 20 cm (d) 8 cm 51. The curved surface area of a right circular cone of height 16 cm and base radius
12 cm is (a) 753.6 cm2 (b) 1205.76 cm2 (c) 863.8 cm2 (d) 907.6 cm2
52. The curved surface area of a right circular cone of slant height 10 cm and base
radius 10.5 cm is (a) 185 cm2 (b) 160 cm2 (c) 165 cm2 (d) 195 cm2
53. The slant height of a cone is 26 cm and base diameter is 20 cm. Its height is
(a) 24 cm (b) 25 cm (c) 23 cm (d) 35 cm 54. The curved surface area of a cone is 308 cm2 and its slant height is 14 cm. The
radius of its base is (a) 8 cm (b) 7 cm (c) 9 cm (d) 12 cm
55. A conical tent is 10 m high and the radius of its base is 24 m. The slant height of
tent is (a) 26 m (b) 28 m (c) 25 m (d) 27 m
56. The slant height and base diameter of a conical tomb are 25 m and 14 m
respectively. The cost of white washing its curved surface at the rate of Rs. 210 per 100 m2 is (a) Rs. 1233 (b) Rs. 1155 (c) Rs. 1388 (d) Rs. 1432
57. A joker’s cap is in the form of cone of base radius 7 cm and height 24 cm. The
area of sheet to make 10 such caps is (a) 5500 cm2 (b) 6500 cm2 (c) 8500 cm2 (d) 3500 cm2
58. A solid right cylinder cone is cut into two parts at the middle of its height by a
plane parallel to its base. The ratio of the volume of the smaller cone to the whole cone is (a) 1 : 2 (b) 1 : 4 (c) 1 : 6 (d) 1 : 8
59. The curved surface area of a sphere of radius 7 cm is:
(a) 516 cm2 (b) 616 cm2 (c) 716 cm2 (d) 880 cm2 60. The curved surface area of a hemisphere of radius 21 cm is:
(a) 2772 cm2 (b) 2564 cm2 (c) 3772 cm2 (d) 4772 cm2 61. The curved surface area of a sphere of radius 14 cm is:
(a) 2464 cm2 (b) 2428 cm2 (c) 2464 cm2 (d) none of these.
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62. The curved surface area of a sphere of diameter 14 cm is: (a) 516 cm2 (b) 616 cm2 (c) 716 cm2 (d) 880 cm2
63. Total surface area of hemisphere of radius 10 cm is
(a) 942 cm2 (b) 940 cm2 (c) 842 cm2 (d) 840 cm2 64. The radius of a spherical balloon increases from 7 cm to 14 cm s air is being
pumped into it. The ratio of surface area of the balloon in the two cases is: (a) 4 : 1 (b) 1 : 4 (c) 3 : 1 (d) 1 : 3
65. A matchbox measures 4 cm x 2.5 cm x 1.5 cm. The volume of packet containing 12 such boxes is: (a) 160 cm3 (b) 180 cm3 (c) 160 cm2 (d) 180 cm2
66. A cuboidal water tank is 6 m long, 5 m wide and 4.5 m deep. How many litre of water can it hold? (a) 1350 liters (b) 13500 liters (c) 135000 liters (d) 135 liters
67. A cuboidal vessel is 10 m long and 8 m wide. How high must it be made to hold 380 cubic metres of a liquid? (a) 4.75 m (b) 7.85 m (c) 4.75 cm (d) none of these
68. The capacity of a cuboidal tank is 50000 litres. The length and depth are respectively 2.5 m and 10 m. Its breadth is (a) 4 m (b) 3 m (c) 2 m (d) 5 m
69. A godown measures 40 m × 25 m × 10 m. Find the maximum number of wooden crates each measuring 1.5 m × 1.25 m × 0.5 m that can be stored in the godown. (a) 18000 (b) 16000 (c) 15000 (d) 14000
70. A river 3 m deep and 40 m wide is flowing at the rate of 2 km per hour. How
much water will fall into the sea in a minute? (a) 4000 m3 (b) 40 m3 (c) 400 m3 (d) 40000 m3
71. The circumference of the base of a cylindrical vessel is 132 cm and its height is 25
cm. How many litres of water can it hold? (a) 33.75 litre (b) 34.65 litre (c) 35.75 litre (d) 38.75 litre
72. If the lateral surface of a cylinder is 94.2 cm2 and its height is 5 cm, then find
radius of its base (a) 5cm (b) 4cm (c) 3cm (d) 6cm
73. It costs Rs 2200 to paint the inner curved surface of a cylindrical vessel 10 m
deep. If the cost of painting is at the rate of Rs 20 per m2, find radius of the base, (a) 1.75 m (b) 1.85 m (c) 1.95 m (d) 1.65 m
74. The height and the slant height of a cone are 21 cm and 28 cm respectively. Find
the volume of the cone. (a) 5546 cm3 (b) 7546 cm3 (c) 5564 m3 (d) 8546 cm3
75. Find the volume of the right circular cone with radius 6 cm, height 7 cm
(a) 254 cm3 (b) 264 cm3 (c) 274 cm2 (d) 284 cm3 76. The radius and height of a conical vessel are 7 cm and 25 cm respectively. Its
capacity in litres is
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(a) 1.232 litre (b) 1.5 litre (c) 1.35 litre (d) 1.6 litre 77. The height of a cone is 15 cm. If its volume is 1570 cm3, find the radius of the
base. (a) 12 cm (b) 10 cm (c) 15 cm (d) 18 cm
78. If the volume of a right circular cone of height 9 cm is 48 cm3, find the diameter of its base. (a) 12 cm (b) 10 cm (c) 6 cm (d) 8 cm
79. A conical pit of top diameter 3.5 m is 12 m deep. What is its capacity in kilolitres?
(a) 38.5 kl (b) 48.5 kl (c) 39.5 kl (d) 47.5 kl
80. Find the capacity in litres of a conical vessel with radius 7 cm, slant height 25 cm (a) 1.232 litre (b) 1.5 litre (c) 1.35 litre (d) none of these
81. The diameter of the moon is approximately one-fourth of the diameter of the earth. What fraction of the volume of the earth is the volume of the moon?
(a) 164
(b) 132
(c) 116
(d) 148
82. The dimensions of a cuboid are 50 cm x 40 cm x 10 cm. Its volume in litres is: (a) 10 litres (b) 12 litres (c) 20 litres (d) 25 litres
83. The volume of a cuboidal tank is 250 m3. If its base area is 50 m2 then depth of the tank is (a) 5 m (b) 200 m (c) 300 m (d) 12500 m
84. The volume of a cuboidal solid of length 8 m and breadth 5 m is 200 m3. Find its
height. (a) 5 m (b) 6 m (c) 15 m (d) 18 m
85. The curved surface area of a sphere is 616 cm2. Its radius is
(a) 7 cm (b) 5 cm (c) 6 cm (d) 8 cm
86. If radius of a sphere is 23d then its volume is
(a) 33281
d (b) 3234
d (c) 3323
d (d) 3343
d
87. The capacity of a cylindrical tank is 6160 cm3. Its base diameter is 28 m. The
depth of this tank is (a) 5 m (b) 10 m (c) 15 m (d) 8 m
88. Base radius of two cylinder are in the ratio 2 : 3 and their heights are in the ratio 5
: 3. The ratio of their volumes is (a) 27 : 20 (b) 25 : 24 (c) 20 : 27 (d) 15 : 20
89. If base radius and height of a cylinder are increased by 100% then its volume increased by: (a) 30% (b) 40% (c) 42% (d) 33.1%
90. The diameter of a sphere is 14 m. The volume of this sphere is
(a) 1437 13
m3 (b) 1357 13
m3 (c) 1437 23
m3 (d) 1337 23
m3
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91. The volume of a sphere is 524 cm3. The diameter of sphere is (a) 5cm (b) 4cm (c) 3cm (d) 7cm
92. The total surface area of a cylinder is 40 cm2. If height is 5.5 cm then its base radius is (a) 5cm (b) 2.5cm (c) 1.5cm (d) 10cm
93. The area of circular base of a right circular cone is 78.5 cm2. If its height is 12 cm then its volume is (a) 31.4 cm3 (b) 3.14 cm3 (c) 314 cm3 (d) none of these
94. The base radius of a cone is 11.3 cm and curved surface area is 355 cm2. Its height
is (Take 355113
)
(a) 5 cm (b) 10 cm (c) 11 cm (d) 9 cm 95. The volume and the surface area of a sphere are numerically equal, then the radius
of sphere is (A) 0 units (B) 1 units (C) 2 units (D) 3 units
96. A cylinder, a cone and a hemisphere are of the same base and of the same height.
The ratio of their volumes is (A) 1 : 2 : 3 (B) 2 : 1 : 3 (C) 3 : 1 : 2 (D) 3 : 2 : 1
97. Small spheres, each of radius 2cm, are made by melting a solid iron ball of radius
6cm, then the total number of small spheres is (A) 9 (B) 6 (C) 27 (D) 81
98. Three solid spheres of diameters 6cm, 8cm and 10cm are melted to form a single
solid sphere. The diameter of the new sphere is (A) 6 cm (B) 4.5 cm (C) 3 cm (D) 12 cm
99. The radii of the ends of a frustum of a cone 40 cm high are 38 cm and 8 cm. The
slant height of the frustum of cone is (A) 50 cm (B) 10 7 cm (C) 60.96 cm (D) 4 2 cm
100. The circular ends of a bucket are of radii 35 cm and 14 cm and the height of
the bucket is 40 cm. Its volume is (A) 60060 cm3 (B) 80080 cm3 (C) 70040 cm3 (D) 80160 cm3
101. If the radii of the ends of a bucket are 5 cm and 15 cm and it is 24 cm high,
then its surface area is (A) 1815.3 cm2 (B) 1711.3 cm2 (C) 2025.3 cm2 (D) 2360 cm2
102. If the radii of the ends of a 42 cm high bucket are 16 cm and 11 cm, determine
its capacity (take 22π7
)
(A) 24222 cm3 (B) 24332 cm3 (C) 24322 cm3 (D) none of these
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PROBABILITY
IMPORTANT FACTS AND FORMULA 1. Experiment : An operation which can produce some well-defined outcome is called an experiment 2.Random Experiment: An experiment in which all possible outcome are known and the exact output cannot be predicted in advance is called an random experiment Eg of performing random experiment: (i) Rolling an unbiased dice (ii)Tossing a fair coin (iii)Drawing a card from a pack of well shuffled card (iv)Picking up a ball of certain color from a bag containing ball of different colors Details: (i)When we throw a coin. Then either a head(h) or a tail (t) appears. (ii)A dice is a solid cube, having 6 faces ,marked 1,2,3,4,5,6 respectively when we throw a die , the outcome is the number that appear on its top face . (iii)A pack of cards has 52 cards it has 13 cards of each suit ,namely spades, clubs ,hearts and diamonds Cards of spades and clubs are black cards Cards of hearts and diamonds are red cards There are 4 honors of each suit These are aces ,king ,queen and jack King, Queen and Jack are called face cards 3.Sample space : When we perform an experiment ,then the set s of all possible outcome is called the sample space Eg of sample space: (i)In tossing a coin ,S={H,T} (ii)If two coin are tossed ,then S={HH ,TT, HT, TH}. (iii)In rolling a die we have,s={1,2,3,4,5,6}. 4.Event:any subset of a sample space. 5. Probability of occurrence of an event. Let S be the sample space and E be the event . Then, P(E)=n(E)/n(S). 6. Results on probability: (i) P(sure event) = 1 (ii) 0< P(E)<1 (iii) P(impossible event)=0 (iv) Ffor any event a and b, we have:
P(AB)=P(A)+P(B)-P(AB)
(v)If A denotes (not-A),then P(A)=1 – P( A ). MCQ QUESTIONS FOR PRACTICE 1. There are 6 marbles in a box with number 1 to6 marked on each of them . What is
the probability of drawing a marble with number 2 ?
(a) 16
(b) 15
(c) 13
(d) 1
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2. A die is thrown once . What will be the probability of getting a prime number ?
(a) 16
(b) 12
(c) 1 (d) 0
Cards are marked with numbers 1 to 25 are placed in the box and mixed thoroughly. One card is drawn at random from the box. Answer the following questions (Q3-Q12)
3. What is the probability of getting a number 5?
(a) 1 (b) 0 (c) 125
(d) 15
4. What is the probability of getting a number less than 11?
(a) 1 (b) 0 (c) 15
(d) 25
5. What is the probability of getting a number greater than 25?
(a) 1 (b) 0 (c) 15
(d) 25
6. What is the probability of getting a multiple of 5?
(a) 1 (b) 0 (c) 125
(d) 15
7. What is the probability of getting an even number?
(a) 1 (b) 0 (c) 1225
(d) 1325
8. What is the probability of getting an odd number?
(a) 1 (b) 0 (c) 1225
(d) 1325
9. What is the probability of getting a prime number?
(a) 825
(b) 925
(c) 1225
(d) 1325
10. What is the probability of getting a number divisible by 3?
(a) 825
(b) 925
(c) 1225
(d) 1325
11. What is the probability of getting a number divisible by 4?
(a) 825
(b) 925
(c) 625
(d) 325
12. What is the probability of getting a number divisible by 7?
(a) 825
(b) 925
(c) 625
(d) 325
13. A bag has 4 red balls and 2 yellow balls. A ball is drawn from the bag without
looking into the bag. What is probability of getting a red ball?
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(a) 16
(b) 23
(c) 13
(d) 1
14. A bag has 4 red balls and 2 yellow balls. A ball is drawn from the bag without
looking into the bag. What is probability of getting a yellow ball?
(a) 16
(b) 23
(c) 13
(d) 1
A box contains 3 blue, 2 white, and 5 red marbles. If a marble is drawn at random from the box, then answer the questions from 15 to 19. 15. What is the probability that the marble will be white?
(a) 16
(b) 15
(c) 13
(d) 1
16. What is the probability that the marble will be red?
(a) 16
(b) 12
(c) 1 (d) 0
17. What is the probability that the marble will be blue?
(a) 310
(b) 12
(c) 1 (d) 0
18. What is the probability that the marble will be any one colour?
(a) 16
(b) 12
(c) 1 (d) 0
19. What is the probability that the marble will be red or blue?
(a) 1 (b) 45
(c) 15
(d) 25
A die is thrown once, then answer the questions from 20 to 24. 20. Find the probability of getting a prime number
(a) 16
(b) 12
(c) 1 (d) 0
21. Find the probability of getting a number lying between 2 and 6
(a) 16
(b) 12
(c) 1 (d) 0
22. Find the probability of getting an odd number.
(a) 16
(b) 12
(c) 1 (d) 0
23. Find the probability of getting an even number.
(a) 16
(b) 12
(c) 1 (d) 0
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24. Find the probability of getting a number greater than 4.
(a) 16
(b) 23
(c) 13
(d) 1
A box contains 5 red marbles, 6 white marbles and 4 green marbles. If a marble is drawn at random from the box, then answer the questions from 25 to 30. 25. What is the probability that the marble will be white?
(a) 16
(b) 23
(c) 13
(d) 1
26. What is the probability that the marble will be red?
(a) 16
(b) 23
(c) 13
(d) 1
27. What is the probability that the marble will be green?
(a) 0.3 (b) 12
(c) 1 (d) none of these
28. What is the probability that the marble will be any one colour?
(a) 16
(b) 12
(c) 1 (d) 0
29. What is the probability that the marble will be red or green?
(a) 25
(b) 325
(c) 15
(d) none of these
30. What is the probability that the marble will be blue?
(a) 16
(b) 12
(c) 1 (d) 0
Cards are marked with numbers 1 to 50 are placed in the box and mixed thoroughly. One card is drawn at random from the box. Answer the following questions from 31 to 40. 31. What is the probability of getting a number 5?
(a) 1 (b) 0 (c) 125
(d) 15
32. What is the probability of getting a number less than 11?
(a) 1 (b) 0 (c) 15
(d) 25
33. What is the probability of getting a number greater than 50?
(a) 1 (b) 0 (c) 15
(d) 25
34. What is the probability of getting a multiple of 5?
(a) 1 (b) 0 (c) 125
(d) 15
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35. What is the probability of getting an even number?
(a) 1 (b) 12
(c) 1225
(d) 1325
36. What is the probability of getting an odd number?
(a) 1 (b) 12
(c) 1225
(d) 1325
37. What is the probability of getting a prime number?
(a) 1 (b) 12
(c) 410
(d) 310
38. What is the probability of getting a number divisible by 3?
(a) 825
(b) 925
(c) 1225
(d) 1325
39. What is the probability of getting a number divisible by 4?
(a) 825
(b) 925
(c) 625
(d) 325
40. What is the probability of getting a number divisible by 7?
(a) 825
(b) 925
(c) 625
(d) 325
TABULATION AND GRAPH Tabulation: Tips for solving tables and graphs problems - (1)= Read and view tables and diagram properly. (2)= Put proper attention, what sum rows and sum of columns represents. (3)= Take care regarding of units. (4)= Try to understand the question, sometimes you can solve in your mind by just looking at data. (5)= Most of the questions can be solved by approximation, thus you can save time by avoiding calculation. Bar Graph: This section comprises of questions in which the data collected in a particular discipline are represented in the form of vertical or horizontal bars drawn by selecting a particular scale.one of the parameters is plotted on the horizontal axis and the other on the vertical axis . the candidate is required to understand the given information and thereafetr answer the given questions on the basis of data analysis. Pie-Graph: The pie-chart or a pie-graph is a method of representing a given numerical data in the form of sectors of a circle. The sectors of the circle are constructed in such a way that the area of each sector is proportional to the corresponding value of the component of the data. From geometry, we know that the area of a circle is proportional to the central angle.
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So, the central angle of each sector must be proportional to the corresponding value of the component. Since the sum of all the central angle is 360°, we have
0Value of the componentCentral angle of the component = 360Total value
Line graph:This section comprises of question in which the data collected in a particular discipline are represented by specific points together by straight lines. The points are plotted on a two-dimensional plane taking one parameter on the horizontal axis and the other on the vertical axis. The candidate is required to analyse the given information and thereafter answer the given questions on the basis of the analysis of data. Following bar graph shows marks obtained by a student in 2005–06 and 2006–07 subject wise. Read and answer the questions from Q1 – Q10
1. In which subject has the performance improved the most?
(a) Maths (b) Science (c) S. Science (d) none of these 2. In which subject has the performance deteriorated?
(a) English (b) Science (c) S. Science (d) none of these 3. In which subject is the performance at par?
(a) Hindi (b) Science (c) S. Science (d) none of these 4. Find the marks obtained in Maths by a student in 2005–06 ?
(a) 30 (b) 40 (c) 50 (d) 60 5. Find the marks obtained in Maths by a student in 2006–07 ?
(a) 30 (b) 40 (c) 50 (d) 60 6. Find the marks obtained in Hindi by a student in 2005–06 ?
(a) 30 (b) 40 (c) 50 (d) 60 7. Find the marks obtained in Hindi by a student in 2006–07 ?
(a) 30 (b) 40 (c) 50 (d) 60 8. Find the marks obtained in S. Science by a student in 2005–06 ?
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(a) 30 (b) 40 (c) 50 (d) 60 9. Find the total marks obtained by a student in 2005–06?
(a) 230 (b) 235 (c) 240 (d) none of these 10. Find the total marks obtained by a student in 2006–07?
(a) 230 (b) 270 (c) 240 (d) none of these
Frequency Distribution of Daily Income of 550 workers of a factory is given below. Study the following frequency distribution table and answer the questions from Q1 – Q10.
Class Interval (Daily income in rupees)
Frequency (Number of workers)
100 – 125 45 125 – 150 25 150 – 175 55 175 – 200 125 200 – 225 140 225 – 250 55 250 – 275 35 275 – 300 50 300 – 325 20
Total 550
1. What is the size of class intervals ? (a) 24 (b) 25 (c) 26 (d) 15
2. Which class has the highest frequency ? (a) 200-225 (b) 300-325 (c) 175-200 (d) 150-175
3. Which class has the lowest frequency ? (a) 100-125 (b) 300-325 (c) 175-200 (d) 150-175
4. What is the upper limit of the class interval 250-275? (a) 250 (b) 275 (c) 25 (d) 525
5. Which two classes have the same frequency? (a) III & IV (b) I & II (c) II & V (d) V & VI
6. What is the range of the all class interval? (a) 250 (b) 275 (c) 225 (d) 525
7. What is the lower limit of the class interval 250-275? (a) 250 (b) 275 (c) 25 (d) 525
8. What is the total number of workers having daily income less than 250? (a) 300 (b) 445 (c) 305 (d) 550
9. What is the total number of workers having daily income more than 200? (a) 300 (b) 445 (c) 305 (d) 550
10. What is the total number of workers having daily income between 150–250? (a) 300 (b) 445 (c) 375 (d) 550
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The number of hours for which students of particular class watched television during holidays is shown through the graph given below. See and answer the questions from Q1 – Q5.
1. For how many hours did the maximum number of students watch TV ?
(a) 4-5 hrs (b) 6-7 hrs (c) 3-4 hrs (d) 2-3hrs
2. How many students watched TV for less than 4 hrs ? (a) 12 (b) 34 (c) 4 (d) 8
3. How many students spent more than 5 hrs in TV watching ? (a) 14 (b) 0 (c) 6 (d) 8
4. For how many hours did the minimum number of students watch TV ? (a) 2-3 hrs (b) 6-7 hrs (c) 1-2 hrs (d) 3-4hrs
5. How many students spent less than 5 hrs in TV watching ? (a) 34 (b) 32 (c) 8 (d) 66
Adjoining pie-chart gives the expenditure (in %age) on various items and savings of a family during a month. Study the given pie-chart and answer the questions from Q6 – Q10.
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6. On which item the expenditure was maximum ? (a) food (b) education (c) others (d) transport
7. On which item the expenditure was minimum ? (a) food (b) education (c) others (d) transport
8. Expenditure on which item is equal to total savings of the family ?
(a) food (b) education (c) others (d) transport
9. Expenditure on which item is equal to total savings of the House Rent? (a) food (b) education (c) clothes (d) transport
If the monthly savings of the family is Rs 3000, 10. What is the monthly income of the family?
(a) 30000 (b) 20000 (c) 25000 (d) 40000
11. What is the monthly expenditure on cloths ? (a) 3000 (b) 2000 (c) 2500 (d) 1000
12. What is the monthly expenditure on education for children ?
(a) 3000 (b) 2000 (c) 2500 (d) 1000 13. What is the monthly expenditure on education for others?
(a) 3000 (b) 2000 (c) 2500 (d) 4000 14. What is the monthly expenditure on education for Transport?
(a) 3000 (b) 2000 (c) 2500 (d) 1000 15. What is the monthly expenditure on education for Food?
(a) 3000 (b) 5000 (c) 2500 (d) 4000
Ex.1 Production of steel by six different companies in three cosecutive years 1994-95-96( In Lakh Tonnes ) are being given
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Q(1) What is the difference between average production of the six companies in 1995 and average production of the same companies in 1994 ? Solution : Sum of production in 1995= 55+55+40+70+70+45=335 Sum of production in 1994=50+45+30+50+70+35=280
335 280, 9.166666676
So difference
Difference=9.16666 tonnes.
Q(2) What is the % decline in production by company C from 1995 to 1996 ? Solution :For company c production in 1995= 40 and in 1996=35
40 35% 100 12.5%40
Decline
Q(3) Which of the following companies recorded the minimum % growth from 1994 to 1995 ? (a) A (b) B (c) C (d) D (e) F Solution :On seeing graph it is clear that company E has recorded minimum growth from 1994 to 1995
Q(4) Production of company ' C ' in 1995 and production of company ' F ' in 1994 together is what % of production of B in 1996 ? Solution : Production of C in 1995 = 40 , Production of F in 1994 = 35 Sum = 75, production of B in 1996 = 60
75% 100 125%60
Q(5) In which of the following pairs of companies the difference between average production for the three years is maximum ? (a) E and F (b) D and F (c) E and C (d) A and E (e) None of these Solution : From the graph we can see that highest production average is for company E and Lowest production average is for company C, so difference between average production for the three years is maximum for company E and C, So, option (c) is correct
Ex.2 Total sale of English and Hindi Newspapers in Five different Localities of a city are given. ( Note : Do not get confused by axis of data, it is shown in cylindrical way, so value is same as shown in y axis )
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Q(1) What is the difference between the total sale of english Newspapers and total sale of Hindi newspapers in all the localities together ? (a) 6000 (b) 6500 (c) 70000 (4)7500 (5) None of these
Solution : Total Sum of all the English News papers = 7500 + 9000 + 9500 + 7000 + 6500 = 39500 Total Sum of all the Hindi News papers = 5500 + 8500 + 4500 + 9500 + 5000 = 33000 Difference = 39500 - 33000 = 6500, so option (b) is correct Q(2) The sale of English News paper in locality A is approximatley what % of the total sale of english newspapers in all the localities together ? (a) 527 (b) 25 (c) 111 (4)236 (5) 19 Solution : Total sum of sale of english newspapers in all the localities together = 39500 as calculated above Sale of English News paper in locality A = 7500 ( from fig. above )
7500% 500 1939500
Required
So option (5) is correct. Q(3) What is the respective ratio of the sale of Hindi Newspapers in locality A to the sale of Hindi Newspapers in locality D (a) 11 : 19 (b) 6 : 5 (c) 5 : 6 (4)19 : 11 (5) None of these Solution :sale of Hindi Newspapers in locality A=5500 sale of Hindi Newspapers in locality D=9500
5500 119500 19
Required ratio
So, option (a) is correct Q(4) The sale of English Newspaper in localities B and D together is approximately what % of the sale English Newspaper in localities A, C and E together ? (a) 162 (b) 84 (c) 68 (4)121 (5) 147 Solution : Sum of the sale of English Newspaper in localities B and D together = 9000 + 7000 = 16000 Sum of the sale of English Newspaper in localities A, C and E together = 7500 + 9500 + 6500 = 23500
16000% 100 68%23500
Required
So, option (c) is correct Q(5) What is the average sale of hindi news papers in all the localities together ? (a) 6600 (b) 8250 (c) 5500 (4)4715 (5) None of these Solution : Sum of the sale of Hindi Newspapers in all localities = 33000 as calculated in Q(1)
33000 66005
Average
So, option (a) correct Ex.3 - The number of students studying in different faculties in the years 2001 and 2002 from state A is as follows
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Ex. In which faculty there was decrease in the number of students from 2001 to 2002 ? (a) Arts (b)Agriculture (c) Pharmacy (d)None (e) None of these Solution : For Arts, Number of students in 2001 and 2002
12 1135000 4200, 40000 4400100 100
For Agriculture Number of students in 2001 and 2002 7 535000 2450, 40000 2000
100 100
For pharmacy % is already more and total number of students are already more in 2002, so correct option will be for Agriculture, option (b) is true Ex. What is the ratio between the number of students studying pharmacy in the years 2001 and 2002 respectively ? (a) 4 : 3 (b) 3 : 2 (c) 2 : 3 (d) 7 : 12 (e) None of these
35000
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Solution : Ratio between the number of students studying pharmacy in the years 2001 and 2002
=
635000 71009 1240000
100
Option (d) is true Ex. What was the approximate percentage increase in the number of students of Engineering from the year 2001 to 2002 ? (a) 17 (b) 15 (c) 25 (d) 23 (e) 20 Solution : Number of engineering students in 2001 =
1835000 6300 2001100
in
Number of engineering students in 2002 = 1940000 7600 2002100
in
Total increase = 7600 - 6300 = 1300
1300% 100 20.63% 20%6300
Increase
Ex. In the year 2001, the number of students studying Arts and Commerce together is what percentage of the number of students studying these subjects together in 2002 ? (a) 76 (b) 85 (c) 82 (d) 79 (e) None of these Solution : Number of students studying Arts and Commerce together in 2001 =
3435000 11900 2001100
in
Number of students studying Arts and Commerce together in 2002 =
3540000 14000 2002100
in
11900% 100 85%14000
Required
So, option (b) is true
Ex. In which of the following faculties the percent increase in the number of students was minimum from 2001 to 2002 ? (a) Arts (b) Science (c) Commerce (d) Medicine (e) Engineering Solution : This question is quit lengthy, First find respective number of students in the given years according to subjects
Arts Science Commerce Medicine Engineering
2001 4200 8400 7700 3850 6300 2002 4400 8800 9600 4000 7600
4400 4200: 100 4.761%4200
For arts
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8800 8400: 100 4.761%8400
For Science
9600 7700: 100 24.67%7700
For Commerce
4000 3850: 100 3.9%3850
For Medicine
7600 6300: 100 20.63%6300
For Engineering
Ex. Study the given graph and table and answer the following questions given below. ( Total population of all states given in Pie chart is 25 lakhs). In the year 1998 the data of different states regarding population of states
Sex and literacy wise population ratio
Sex Literacy States M F Literate Illiterate
U P 5 3 2 7
Bihar 3 1 1 4
A P 2 3 2 1
Kar 3 5 3 2
MH 3 4 5 1
T N 3 3 7 2
Kerala 3 4 9 4 Q(1) Approximately what is the total number of literate people in MH and Kar together ? (a) 4.5 lakhs (b) 6.5 Lakhs (c) 3 lakh (d) 3.5 lakhs (e) 6 lakhs Solution : Population of Kar is 15 %
25 15 15Population of Kar =100 4
lakhs
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So, Total number of literate people in Kar is :
3 15Literates of Kar = 2.253+2 4
lakhs
Population of MH is 11 %
25 11 11Population of MH =100 4
lakhs
5 11Literates of MH = 2.295+1 4
lakhs
So, number of literate people in MH and Kar together = 2.29 + 2.25 = 4.54 lakhs So, option (a) is correct Q(2). Approximately what will be the percentage of total male in UP , MH and kerala of the total population of the given states ? (a) 20% (b) 18% (c) 28% (d) 30% (e) 25% Solution : Population in UP is 25 %
25 25 25Population of UP = 6.25100 4
lakhs
5 25Male people in UP = 3.935+3 4
lakhs
Population in MH = 2.75 lakhs from Q(1)
3 11Male people in MH = 1.173+4 4
lakhs
Population in Kerala = 8 %
25 8Population of Kerala = 2100
lakhs
3Male people in Kerala = 2 0.853+4
lakhs
So, Sum of male population of UP, MH and kerala = 3.93 + 1.17 + 0.85 = 5.95 lakhs
255.96 24% 25%100
x x of total
option (e) correct Ex. Number of hotels in a state, according to years are given ( Study the given chart carefully and then answer the questions accordingly ) Q(1) The approximate % increase in hotels from year 1989 to 1994 was (a) 75 (b) 100 (c) 125 (d) 150 (e) 175 Solution :
838 410% 100 104.3 100%410
increase
Option (b) is correct
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Q(2) If the number of newly made hotels in 1991 was less by 10 then what is the ratio of the number of hotels in 1991 and that in 1990 ? (a) 14 : 11 (b) 3 : 4 (c) 4 : 5 (d) 5 : 4 (e) 1 : 4 Solution : No of hotels in 1991 will be 570 - 10 = 560
560 14%410 11
Required
Option (a) is correct Q(3) If the % increase in the number of hotels from 1993 to 1994 continued up to 1995 then what is the number of hotels built in 1995 ? (a) Minimum 75 (b) Minimum 70 (c) Minimum 50 (d) Minimum 139 (e) Minimum 80 Solution : First find % increase in 1994 from 1993
838 710% 100 18%710
increase
Now 18 % of 838 = 150 So, nearest option is (d) Minimum 139 hotels Q(4) In which of the given years increase in hotels in comparison to the previous year is the maximum ? (a) 1990 (b) 1991 (c) 1992 (d) 1993 (e) 1994 Solution:
440 4101990 100 7.31%410
In
570 4401991 100 29.54%
440In
710 5701992 100 24.56%
570In
710 7101993 100 0%
710In
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838 7101994 100 18.05%710
In
So option (b) is correct Q(5) If increase in hotels from 1991 to 1992 is P % and increase in hotels from 1992 to 1994 is Q % , then which of the following relations between P and Q is true ? (a) Data is inadequate (b) P < Q (c) P = Q (d) P > Q (e) None of these Solution : P % = 24.56 % from question (4) Q % = 18.05 % So P > Q, option (d) correct Ex. % profit earned by two companies over the years is given in graph. Also
% 100Income ExpenditureProfitExpenditure
Ex. If the expenditure of Company B in 2000 was Rs. 200 crores, what was its income ? (a) Rs. 240 crores (b) Rs. 220 crores (c) Rs. 160 crores (d) Can not be determined (e) None of these. Solution : Let income be Rs x crores so, we can use the above formula as
20020 100200
x
40 = x - 200 x= 240 crores Ex. If the income of company A in 2002 was Rs. 600 crores. What was its expenditure ? (a) Rs. 360 crores (b) Rs. 480 crores (c) Rs. 375 crores (d) Can not be determined (e) None of these. Solution : For company A in 2002 % profit was 60 % , Let expenditure be x crores , so
60060 100 375x x croresx
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Ex. If the income of a company B in 1998 was Rs. 200 crores, what was its profit in 1999 ? (a) Rs. 21.5 crores (b) Rs. 153 crores (c) Rs. 46.15 crores (d) Can not be determined (e) None of these. Solution : Profit can be calculated only when Income and expenditure of the given year should be known. So, option (d) is correct Ex. If the Incomes of the two companies in 1998 were equal, what was the ratio of their Expenditure ? (a) 1 : 2 (b) 26 : 27 (c) 100 : 67 (d) Can not be determined (e) None of these. Solution : Let, the income of both companies be P, expenditure of A is E1 and expenditure of B is E2 Now we can write,
11
1
10035135
P E PEE
22
2
10030130
P E PEE
1
2
130 26135 27
EE
So, option (b) is correct Ex. What is the % increase in % profit for company B from year 2000 to 2001 ? (a) 75 (b) 175 (c) 42.86 (d) Can not be determined (e) None of these.
Solution : 35 20% 100 75%20
increase
Ex. Study the table and answers the questions.
Financial Statement of A company Over the years (Rupees in Lakhs )
Year
Gross Turnover
Profit before interest and depreciation
Interest Rs.
Deprciation Rs.
Net profit Rs.
2000-01 1360 381.9 300 70 10.66 2001-02 1402 403.92 315.35 71.1 18.45 2002-03 1538.4 520.05 390.8 80.01 49.15 2003-04 2116.34 600.02 440.89 89 66 2005-06 2521 810 500.9 91.92 212.8 2006-07 2758.99 920 600 99 220.8
Q (1) During which year did the 'Net Profit ' exceed Rs. 1 crore for the first time ? (a) 2006-07 (b)2005-06 (c) 2003-04 (d) 2002-03 (e) None of these Solution :By looking at the table we find option (b) as correct Q(2) During which year was the " Gross Turnover " closet to thrice the 'Profit before
Interest and Depreciation' ? (a) 2006-07 (b)2005-06 (c) 2003-04 (d) 2002-03 (e) 2001-02 Solution : Here we have to find out the ratio of "Gross turnover " to the "Profit before
Interest and Depreciation " for 2006-07 ratio = 2758.99 / 920 = 3.0 , since we get answer by hit and trial of first
option only then we need not have to find other option. So, correct option is (a) 2006-07
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Q(3) During which of the given year did the 'Net profit ' form the highest proportion of the ' Profit before Interest and Depreciation ?
(a) 2005-06 (b) 2003-04 (c) 2002-03 (d) 2001-02 (e) 2000-01 Solution : Here we will find the reverse ths ratio between ( Profit before Interest and
Depreciation / Net Profit ) and try to find lowest ratio also we use approximation for saving time
For 2005-06, 81 / 21 = 4 approx. For 2003-04, 60 / 6.6 = 9 approx For 2002-03, 520 / 50 = 10 approx For 2001-02, 40 / 1.8 = 22 approx For 2000-01, 380 / 10 = 38 approx So, lowest ratio is for 2005-06 it means reverse ratio between (Net Profit / Profit before
Interest and Depreciation ) will be highest, so option (a) is correct Q(4) Which of the following registered the lowest increase in terms of rupees from the
year 2005-06 to the year 2006-07 ? (a) Gross turnover (b) Profit before interest and depreciation (c) Depreciation (d) Interest (e) Net Profit Solution : We calculation with approximation in mind (a) 2758 - 2520= 2200 approx.
(b) 92 -81 =11= 110 approx
(c) 99 - 91.92 = 08.09 approx
(d) 600 - 505 = 95 approx
(e) 220 - 212= 08 approx
So correct option is (e) Net profit
Q(5) The gross Turnover for 2002-03 is of what percentage of the 'Gross Turnover ' for
2005-06 ? (a) 61 (b) 163 (c) 0.611 (d) 39 (e) 0.006 Solution : 2002-03 Gross Turnover 1538.40 For 2005-06 Gross Turnover 2521.00 So option (a) is correct Ex. Marks otained by different students in different subjects
Subject (Maximum Marks )
Hindi English Maths Social std. Science Sanskrit Phy. Edu Students
100 100 100 100 75 50 75 Anupam 85 95 87 87 65 35 71
Bimal 72 97 55 77 62 41 64 Chaman 64 78 74 63 55 25 53
Devendar 65 62 69 81 70 40 50 Girish 92 82 81 79 49 30 61 Vivek 55 70 65 69 44 28 30
Q(1) How many students have scored the lowest marks in two or more subjects ? (a) 2 (b) 3 (c) 1 (d) 0 (e)4
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Solution :Here option (a) is correct by looking at table we find Chaman and Vivek has scored lowest marks in two or more subjects
Vivek in Hindi, Science, Phy. Edu Chaman in Social std, Sanskrit Q(2) Who has scored the highest marks in all the subjects together ? (a) Devendar (b) Chaman (c) Anupam (d) Girish (e)Bimal Solution : For this calculate total toatal of each students in options
Devendar = 65 + 62 + 69 + 81 + 70 + 40 + 50 = 437
Chaman = 64 + 78 + 74 + 63 + 55 + 25 + 53 = 412
Anupam = 85 + 95 + 87 + 87 + 65 + 35 + 71 = 525
Girish = 92 + 82 + 81 + 79 + 49 + 30 + 61= 474
Bimal = 72 + 97 + 55 + 77 + 62 + 41 + 64 = 468
So correct option is (3) Anupam
Q (3) What is the percentage of Devendar's marks (upto two digits after decimal) in all
the subjects together ? (a) 88.63 (b) 77.38 (c) 67.83 (d) 62.83 (e)72.83 Solution : Devendar's total marks = 437
437Required % = 100 72.83600
Q (4) Marks obtained by Chaman in Hindi are what percentage of marks ( upto two digits after decimal ) obtained by Anupam in the same subject ?
(a) 75.92 (b) 78.38 (c) 77.29 (d) 75.29 (e)72.83 Solution: Marks obtained in Hindi by Chaman = 64, by Anupam= 85
64Required % = 100 75.2985
Q (5) What are the average marks obtained by all the students together in Science ?
(a) 55.75 (b) 57.50 (c) 60.00 (d) 59.50 (e)58.00 Solution : Average maks obtained in Science =
65+62+55+70+49+44 345Average = 57.56 6
So, option (b) true
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PRACTICE PAPER (QUANTITATIVE APTITUDE)
1. If a pipe A fills a tank in 10 hours and a pipe B fills it in 15 hours, then the tank will be filled by A and B together in how much time?
A. 10 hrs. B. 9 hrs C. 6 hrs D. 8 hrs 2. Two pipes P & Q can separately fill a cistern in 12 minutes and 18 minutes
respectively. In how many minutes can both the pipes fill the cistern, if opened together?
A. 7 1/5 min B. 7 min C. 7 ½ min D. 10 min 3. Two piped A & B can fill a tank in 36 hrs and 45 hrs respectively. If both pipes are
opened simultaneously, how much time will be taken to fill the tank? A. 20 hrs B. 21 hrs C. 22 hrs D. None 4. A pipe can fill a tank in 15hrs. Due to leak in the bottom, it is filled in 20 hrs. If the
tank is full, how much time will the leak take to empty it? A. 55 hrs B. 60 hrs C. 61 hrs D. None 5. If a pipe A fills a tank in 10 hrs and pipe B empties it in 15 15 hrs, then in how
much time will the tank be filled up if A & B are both opened? A. 4 hrs B. 14 hrs C. 20 hrs D. 30 hrs 6. Two taps A & B can fill a water reservoir in 12 and 15 hrs respectively. How long
would the two taps take to fill this reservoir if both are opened together? A. 6 ¼ hrs B. 6 ½ hrs C. 7 1/3 hrs D. None 7. Two taps can fill a tank in 24 minutes and 30 minutes respectively. Both of them
are opened together, but the first tap is turned off after 8 minutes. Now find how long would the second tap take to fill the tank.
A. 24 ½ min B. 12 min C. 13 2/3 min D. 16 min 8. Two pipes A & B can fill a tank in 24 min and 32 min respectively. If both the
pipes are opened simultaneously, the time after which B should be closed so that tank is filled in 18 minutes would be?
A. 18 min B. 10 min C. 9 min D. 8 min 9. Two pipes A & B can fill a tank in 24 minutes and 36 minutes respectively. If both
the pipes are opened simultaneously, the time after which A should be closed so that tank is filled in 18 min would be?
A. 12 min B. 10 min C. 9 min D. None 10. Two pipes A & B separately fill a cistern in 12 and 15 min respectively while a 3rd
pipe C can empty it in 10 minutes. How long will it take to fill the cistern if all pipes are opened?
A. 10 min B. 20 min C. 30 min D. 22 min 11. A pipes fills a tank in 2 hrs and another files the tank in 3hrs but a 3rd pipe empties
the filled tank in 5 hrs. Then if three pipes are opened, the tank will be filled in: A. 1 hr B. 1 11/19 hr C. 2 1/9 D. None
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12. What is the value of: 11211 11
4
A. 3/2 B. 5/4 C. 5/6 D. 1
1. 342 729 2536 9 196
A. 135/14 B. 18/5 C. 18/7 D. None 14. 9.75 + 25.88 + x = 41.18 A. 5.55 B. 5.75 C. 6.57 D. 4.23
15. 30% of 270 + 58
of 64 = x
A. 120 B. 81 C. 121 D. 242 16. 73.85 + 215.345 – 167.2134 = x A. 456.4084 B. 121.6711 C. 120.8296 D. None 17. x% of 150 + 250 = 280 A. 30 B. 10 C. 40 D. 20 18. 25% of 40 ÷4% of 25 = x A. 1 B. 10 C. 0 D. 2 19. Simplify: (34 * 37) ÷ 310
A. 3 B. 9 C. 27 D. 81
20. Find the value of: 2.70 2.70 4.30 4.30 8.60 2.702.70 4.30
A. 6.8 B. 7.6 C. 7.0 D. 8.5
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PRACTICE PAPER (QUANTITATIVE APTITUDE)
Directions (Q. 1 – 10): what should come in place of question mark (?) in the following questions? 1. (1331)2 ÷ (14641)3 = (11)? × (121) (a) 2 (b) 4 (c) 6 (d) 8 (e) 10 2. 637.28 – 781.47 + 257.39 = ? (a) 113.20 (b) 104.30 (c) 122.40 (d) 133.50 (e) None of these 3. 6% of 350 + 2% of 700 = ?% of 1400 (a) 2 (b) 2.5 (c) 3 (d) 4 (e) None of these 4. 4672 ÷ 40 ÷ 4 = ? (a) 467.2 (b) 29.6 (c) 29.2 (d) 368.8 (e) None of these 5. 7 × ? = 546 ÷ 4 (a) 24.4 (b) 113.5 (c) 37.9 (d) 19.5 (e) None of these 6. 2.4 × (33 ÷ 3.6) = ? (a) 11.6 (b) 16.3 (c) 23.9 (d) 28.5 (e) None of these 7. (5 × 5 × 5 × 5 × 5)5 × (5 × 5 × 5 × 5)6 ÷ (5 × 5 × 5)7 = (5 × 5)? (a) 3 (b) 28 (c) 14 (d) 7 (e) None of these 8. 7.2 × 3.6 × 1.5 = ? (a) 38.88 (b) 36.46 (c) 39.32 (d) 42.18 (e) None of these 9. 13% of 360 + ? = 106 (a) 82.2 (b) 73.6 (c) 61.4 (d) 59.2 (e) None of these 10.39 ÷ 1.3 – 0.7 = ? + 1.6 (a) 13.7 (b) 23.3 (c) 39.3 (d) 27.7 (e) None of these 11.11.8% of 645 = ? (a) 76.17 (b) 78.11 (c) 89.33 (d) 72.13 (e) None of these 12.49×158÷2227=? (a) 959 (b) 353 (c) 151 (d) 1151 (e) None of these 13.2268 – 3376 + 5456 = ? × 80 (a) 112.75 (b) 59.45 (c) 54.35 (d) 68.55 (e) None of these 14.579−623 +829=? (a) 713 (b) 619 (c) 719 (d) 729 (e) None of these
Prepared by: M. S. KumarSwamy, TGT(Maths) Page - 90 -
15.8686 – 6868 = ? × 25 (a) 72.72 (b) 86.86 (c) 68.68 (d) 64.64 (e) None of these 16.4554÷9 + 4792 ÷ 8 = ? (a) 1000 (b) 1054 (c) 1086 (d) 1105 (e) None of these 17.35% of 172of 1260 + 32% of 650 = ? (a) 465.75 (b) 456.25 (c) 466.50 (d) 467.00 (e) None of these 18.2201 ÷ 31 = ? ÷ 9 (a) 629 (b) 639 (c) 649 (d) 659 (e) None of these 19.((4184××2323))+−((8140××1313))= ? (a) 12 (b) 14 (c) 15 (d) 18 (e) None of these 20.166 × 10.5 = ? × 7 (a) 239 (b) 249 (c) 256 (d) 289 (e) None of these 21.The ratio of two numbers is 7 : 5 and the difference between these two
numbers is 16. What is the average of these two numbers? (a) 40 (b) 48 (c) 56 (d) Cannot be determined(e) None of these 22.Which is the next number in the following series? 0, 7, 26, 63, 124, ? (a) 168 (b) 195 (c) 210 (d) 215 (e) None of these 23.In how many different ways can the letters of JAGRAN be arranged? (a) 180 (b) 120 (c) 720 (d) Cannot be determined (e) None of these 24.Five years ago the ratio of the ages of A and B was 2 : 3 and after 7 years it will be 4 : 5. What is the present age of ‘A’? (a) 12 years(b) 15 years(c) 17 years(d) Cannot be determined (e) None of these 25.If an amount of `6, 87, 056 is distributed equally amongst 92 persons, how much amount would each person get? (a) 8128 (b) 6648 (c) 5698 (d) 7486 (e) 7468 26.What would be the simple interest obtained on an amount of `8220 at the rate of 7.5% per annum after six years? (a) Rs. 3, 699 (b) Rs. 3, 798 (c) Rs. 4, 279 (d) Rs. 4, 799 (e) Rs. 2, 681 27.What would be compound interest obtained on an amount of `12600 at the rate of 10% per annum after two years? (a) Rs. 2, 520 (b) Rs. 2, 400 (c) Rs. 2, 600 (d) Rs. 2, 646 (e) Rs. 2, 746
Prepared by: M. S. KumarSwamy, TGT(Maths) Page - 91 -
28.The ratio of the ages of Rajesh to that of Ramesh is 5 : 8. After 10 years the new ratio of their ages will be 7 : 10. What is the age of Ramesh at present? (a) 20 years (b) 25 years (c) 30 years (d) 35 years (e) 40 years 29.A rectangular room has length 24 metre and breadth 21 metre. What will be the total cost if the floor is made at `180 per square metre? (a) Rs. 82, 520 (b) Rs. 82, 820 (c) Rs. 86, 720 (d) Rs. 90, 620 (e) None of these 30.A shopkeeper purchased 25 items at `36 per item. He spent `100 on transportation of these items. What should be the selling price of each item if he wants to make 22.5% as profit? (a) `49 (b) `45 (c) `47 (d) `51 (e) None of these 31.16 men can complete a work in 35 days. In many days can 28 men complete the same piece of work? (a) 18 days (b) 20 days (c) 22 days (d) 16 days (e) None of these 32.The cost of 16 kg of rice is `416. The cost of 14 kg of atta is `294 and the cost of 8 kg of pulse is `352. What will be the total cost of 13 kg rice, 17 kg of atta and 11 kg of pulse? (a) Rs. 1, 069 (b) Rs. 1, 169 (c) Rs. 1, 179 (d) Cannot be determined (e) None of these 33.The population of city is 80, 000. If it increases 11% in the first year and decreases by 9% in the second year, what will be the population of the city after two years? (a) 81, 600 (b) 81, 000 (c) 84, 800 (d) Cannot be determined (e) None of these 34.On selling an item for `96 a man gains 20%. In order to gain 35%. What will be the increase in the selling price? (a) Rs. 10 (b) Rs. 12 (c) Rs. 13 (d) Rs. 15 (e) None of these 35.The cost of 21 chairs and 33 tables is `96, 054. Then what will be the cost of 49 chairs and 77 tables? (a) Rs. 2, 00, 000 (b) Rs. 2, 12, 456 (c) Rs. 2, 24, 126 (d) Data inappropriate (e) None of these 36.The product of two consecutive even numbers is 4, 623. Which is the smaller number? (a) 67 (b) 69 (c) 71 (d) 73 (e) 77
Prepared by: M. S. KumarSwamy, TGT(Maths) Page - 92 -
37.A car travels a distance of 120 km at the rate of 30 km/h. It covers the nest 150 km at the rate of 25 km/h and the last 90 km of its journey at the speed of 18 km/h. what is the average speed of the car? (a) 20 km/h (b) 25 km/h (c) 30 km/h (d) 24 km/h (e) 35 km/h 38.The average age of a man and his son is 42 years. The ratio of their ages 12 years back was 4 : 1. What is the present age of the son? (a) 15 years (b) 16 years (c) 20 years (d) 24 years (e) None of these 39.The sum of money that will give Rs.6 per day (1 year = 365 days) as simple interest at the rate of 12% per annum is (a) Rs. 18, 250 (b) Rs. 20, 360 (c) Rs. 22, 720 (d) Rs. 25, 650 (e) Rs. 27, 900 40.A person spends Rs.2, 325 on his monthly expenditure and the rest 85% of his monthly salary in invested in shares. What is his annual income from salary? (a) Rs. 1, 76, 000 (b) Rs. 1, 80, 000 (c) Rs. 1, 86, 000 (d) Rs. 1, 96, 000 (e) Rs. 2, 40, 000