Quicker Maths - KopyKitab · Ex [3] 2.5 pounds = _____ oz. a. There are 16 oz in a pound so multiply 2.5 by 16 or 40. b. The answer is 40. C. If you need more conversions, most can

Quicker Maths

INDEX

1. Multiplying And Adding Numbers In The Form: ab + bc 1 2. Adding a sequence in the form: 1 + 2 + ... + n 2 3. Approximating - Adding A Series Of Numbers 3 4. Adding And Subtracting Fractions 5 5. Comparing Fractions 6 6. Basic Conversions 8 7. Definitions 11 8. Division 15 9. Double and Half Method 17 10. FOIL Method 18 11. Multiplying Mixed Numbers Whose Fractions Add To 1 20 12. Finding The GCD (Greatest Common Denominator) 22 13. Finding The LCM (Least Common Multiple) 24 14. Multiplying Two Numbers Less Than 100, But Close To 100 25 15. Multiplying Two Numbers Greater Than 100, But Close To 100 26 16. Multiplying by 5 27 17. Multiplying By 11 28 18. Multiplying By 25 29 19. Multiplying By 50 30 20. Multiplying By 101 31 21. Adding a sequence in the form: 1 + 3 + ... + 2n-1 32 22. Multiplying Two Nos Whose Ten’s Digits Are Same & One’s Digits Add To 10 33 23. Order of Operations 34 24. Percents 35 25. prime numbers 39 26. Ratios 40 27. Subtracting Reverses 41 28. Roman Numerals 42 29. Squaring A Number Ending In 5 45 30. Squaring A Number In The Range (40 - 49) 46 31. Squaring A Number In The Range (50 - 59) 47 32. Squaring A Number In The Range (90 - 99) 48 33. Squaring A 2-Digit Number 49 34. Working With Square Roots 50 35. Shortcut - Multiplying by 11 53 36. Shortcut - Multiplying by 12 54 37. Shortcut - Is it divisible by four? 55 38. Shortcut - Decimals Equivalents of Fractions 56 39. Shortcut - Converting Kilos to pounds 58 40. Shortcut - Adding Time 59 41. Shortcut - Temperature Conversions 60 42. Shortcut - Converting Kilometres to Miles 61

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Multiplying And Adding Numbers In The Form: ab + bc

A. From algebra we can factor:

ab + bc = b(a + c)

B. Using numbers instead of variables we get the following:

1. Take out the number that both sides have in common.

2. Add the remaining numbers.

3. Multiply the number in step 1 with the result in step 2 for the answer.

Ex [1] 15 x 12 + 15 x 8 =_________.

a) Rewrite in the form 15 x (12 + 8).

b) 15 x 20 = 300.

c) The answer is 300.

Ex [2] 16 x 16 + 16 x 17 =_________.

a) Rewrite in the form 16 x (16 + 17).

b) 16 x 33 = 48 x 11. .

c) 48 x 11 = 528. .

C. Ex [2] uses a variety of different methods. This is just how I would do the problem,

but there are many different ways of going about solving this problem. This is up to

you.

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Adding a sequence in the form: 1 + 2 + ... + n: A. This sequence is sometimes referred to as Triangular Numbers, and can be solved by

the equation:

∑=

n

ii

1

= 1 + 2 + … + n =2

)1( +⋅ nn

B. Using numbers instead of variables we get the following:

1. Multiply the last number by that number plus 1, then divide by 2.

2. Notice one of these numbers is divisible by 2, so you can divide the even

number by 2 and then multiply by the other number.

Ex [1] 1 + 2 + ... + 10 =_________.

a) From the equation we know this is equal to: 10 x 11/2 or 5 x 11 = 55.

b) The answer is 55.

Ex [2] 1 + 2 + ... + 50 =_________.

a) From the equation we know this is equal to: 50 x 51/2 or

25 x 51 = 1275. See Multiplying by 25.

b) The answer is 1275.

C. Sometimes there might be a number missing to throw you off, so you need to be

careful.

Ex [3] 2 + 3 + 4 + ... + 25 =_________.

a) Notice that the number 1 is missing from the equation. Treat it as though it

were there.

b) From the equation we know this is equal to: 25 x 26/2 or 25 x 13 = 325.

c) Since the number 1 is missing, you should subtract 1 from 325. The

answer is 324.

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Approximating - Adding A Series Of Numbers: A. These types of problems will almost always be found on problem #10 and nowhere

else.

B. Since these problems are approximations you can make it easier and faster on yourself

by rounding some numbers with discretion. A basic rule of thumb is "the larger the

numbers, the more you can round."

Ex [1] 558 + 243 - 132 + 69 = __________.

a. In this problem, the numbers are not large but small, so I would round with

extreme discretion.

b. It is safe to use: 600 + 200 - 130 + 70, because the first number is less than

600 about the same distance as the second number is greater than

100. You would get 740.

c. However, you could use: 560 + 200 - 100 + 70, because 243 is almost the

same distance from 200 as 132 is from 100. You would get 730.

d. The answers can be between 702 and 774.

Ex [2] 4589 + 6743 - 1237 + 555 = _________.

a. In this problem, the numbers are larger, so we have a greater leniency.

b. It is safe to use: 5000 + 6000 - 1000 + 600, because 4589 is close to the

same distance from 5000 as 6743 is from 6000 and also 1237 is close to

1000 but to make sure round 555 up. You would get 10600.

c. The answer can be between 10118 and 11182.

C. Many times there will be numbers on the question that are insignificant and can be

ignored.

Ex [3] 14141 - 1414 - 141 - 14 - 1 = __________.

a. In this problem we are dealing with big numbers and the small ones should

be ignored. The first 2 numbers are the only important ones.

b. It is safe to use: 14400 - 1400, since the numbers are relatively big; we

have more leniency. You would get 13000.

c. The answer can be between 11943 and 13199.

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D. In short, there are numerous ways to going about solving these types of

approximations. It takes practice to learn how much you can round.

E. If you do not feel comfortable rounding so much (as in Ex [3]), then you can round

first and subtract a little off in the end just to be sure. This practice saved me a few

times.

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Adding And Subtracting Fractions:

A. The conventional way of adding and subtracting fractions is by changing the fractions

to have the same denominators, otherwise known as the LCM or Least Common

Multiple (see LCM in Miscellaneous).

Ex [1] __________2

1

3

1 =+ (fraction)

Since1/3 and

1/2 can not be added directly, each fraction needs to be changed

to have a common denominator of 6.

a)1/3 =

2/6 and

1/2 =

3/6.

b)2/6 +

3/6 =

5/6.

c) The answer is 5/6.

B. Another way of adding and subtracting fractions is using the following rule from

algebra (sometimes called cross-multiplication):

db

bcad

d

c

b

a

⋅

+=+

Ex [2] __________6

1

7

2 =−(fraction)

a) 76

7162

6

1

7

2

××−×=−

b) 42

5

42

712

76

7162 =−=×

×−×

c) The answer is 5/42.

C. There are several precautions and suggestions that should be considered when adding

and subtracting fractions.

1. Before using the method in part B, first look to see if one denominator is a

multiple of the other. If it is, the method in part A is faster.

2. Before writing down the answer, make sure the fraction is reduced to its

simplest form.

3. Always know what the question is asking for. Sometimes the answer can be

given in improper fractions, while other times mixed numbers should be

given.

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Comparing Fractions: A. When comparing fractions, we often need to know which fraction is smaller or larger.

b

a

d

c

1. To solve this quickly you can use cross-multiplication to determine which fraction

is smaller or larger:

ad ? bc

b

a

d

c

2. In other words, if ad > bc then the fraction on the left is larger. If ad < bc, then

the fraction on the right is larger. If ad = bc, then the 2 fractions are equivalent.

Ex [1] Which is greater: 6

5or

9

7?

a) Using the rule of cross-multiplication we can compare 9 x 5 and 6 x 7.

b) 9 x 5 = 45.

c) 6 x 7 = 42.

d) Since 45 > 42, the fraction on the left is greater.

e) So the answer is 5/6.

B. Sometimes instead of giving two fractions, the problem will give one fraction and one

decimal. In problems like these, simply change the decimal to a fraction (it does not

have to be in simplest terms) and compare using this method.

Ex [1] Which is smaller: .54 or 11

6?

a) You can change .54 to 54/100 (there is no need to simplify).

b) Using cross-multiplication we can compare 54 x 11 and 6 x 100.

c) 54 x 11 = 594

d) 6 x 100 = 600.

e) Since 594 < 600, the fraction (or in this case the decimal) on the left is

smaller.

f) The answer is .54.

?

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C. In problems like Ex [1] Part B, it would be faster if you knew that 6/11= .5454... Therefore, memorizing the fractions will be useful in situation like

these.

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Basic Conversions:

A. Below is a list of basic conversions that you should know:

Distance:

1000 mm = 1 m 12 in = 1 ft

100 cm = 1 m 3 ft = 1 yd

1000 m = 1 km 5280 ft = 1 mile

1 in = 2.54 cm 1760 yd = 1 mile

Volume:

1000 mL = 1 L 1 pint = 2 cups

1 Tbl = 3 Tsp 2 quarts = 1 pint

1 cup = 16 Tbl 4 quarts = 1 gal

1 cup = 8 oz 231 in3

= 1 gal

Weight:

1000 mg = 1 g 1 lb = 16 oz

1000 g = 1 kg 1 ton = 2000 lbs

B. Examples

Ex [1] 36 yd = ______ in

a. First, convert yd to feet. 36 yd = 108 ft. (Multiply 36 x 3).

b. 108 ft = 12 x 108 in = 1296 in.

c. The answer is 1296.

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Ex [2] How many pints are in 3 gallons? ________

a. First, convert gallons to quarts. 3 gallons = 3 x 4 quarts or 12.

b. Convert quarts to pints: 12 quarts = 12 x 2 pints = 24 pints.


Ex [3] 2.5 pounds = ______ oz.

a. There are 16 oz in a pound so multiply 2.5 by 16 or 40.

b. The answer is 40.

C. If you need more conversions, most can be found on the basic conversion page.

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The following cubes should be memorized (at least through 203):

There are some numbers in which you should memorize even higher powers:

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Definitions:

A. There are many problems on the number sense tests that can be solved by simply

knowing its definition.

Mean - The mean of a list of numbers is simply the average of all the numbers.

Ex [1] Find the mean of 12, 13, 14, 15, and 16.

a. On problems like this, you can add up all the numbers and divide by the

number of terms like:

(12 + 13 + 14 + 15 + 16) y 5 or

b. If each number is being added by a fixed number, you can simply write

the middle number (if the list has an odd number of terms) or you can

take the average of the middle two numbers (if the list has an even

number of terms).

c. For this example, there are 5 terms, so the mean is 14.

d. The answer is 14.

Ex [2] Find the mean of 3, 7, 11, 15, 19, and 23.

a. In this example we can take the average of the middle two numbers: 11

and 15.

b. (11 + 15) y2 is 26y2 or 13.


*NOTE: Many times there is no shortcut and you must add all the terms and

divide by the number of terms.

Median - The median of a set of numbers is simply the middle number. If a set

has an even number of terms, then the median is the average of those

two terms.

*NOTE: The numbers in the set MUST be in sequential or der.

Ex [3] Find the median of 2, 5, 6, 9, 12.

a. Since there are an odd number of terms, the median is the middle

number or 6.

b. The answer is 6.

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Ex [4] Find the median of 2, 2, 4, 4, 1, 2.

a. First, change this set to be {1, 2, 2, 2, 4, 4} mental ly.

b. Since there is an even number of terms we take the average to the

middle terms: 2 and 2.

c. The answer is 2.

Mode - The mode of a set of numbers is simply the number that repeats itself the

most.

Ex [5] Find the mode of 2, 3, 7, 3, 7, 1, 3.

a. In this example there are more 3's than there are of other numbers so

the mode is 3.

b. The answer is 3.

Range - The range of a set of numbers is simply the largest value minus the

smallest value.

Ex [6] The range of 1, 7, -1, 8, 4, and 3 is _______.

a. The largest value is 8 and the smallest value is -1.

b. 8 - (-1) = 9.

c. The answer is 9.

Additive Inverse - The additive inverse of a number is the number that you

would add to make the original number equal to 0.

Ex [7] The additive inverse of -14

5is __________.

a. The number that you would add to make this 0 is 1 4/5.

b. The answer is 1 4/5.

Multiplicative Inverse - The multiplicative inverse of a number is the number

that you would multiply to make the original number equal 1.

Ex [8] The multiplicative inverse of -4/5 is _______.

a. The number that you would multiply to make this 1 is -5/4.

b. The answer is -5/4.

Negative Reciprocal - The negative reciprocal of a number is -1/n. If the number

is a fraction simply flip the fraction and change the sign.

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Ex [9] The negative reciprocal of 4

5is _______.

a. Flipping the fraction and changing the sign we get -5/4.

b. The answer is -5/4.

B. The following definitions are more advanced and therefore, probably will not be seen

on an elementary test.

Geometric Mean - The geometric mean is equated by the following:

1

( ) ni j na a au u u"

1. In other words it is the nth

root of the product of all the numbers in the set.

2. On a number sense test, most of the time you will be dealing with 2

numbers. If you are dealing with 3 or more, each value will have an even

nth

root.

Ex [10] Find the geometric mean of 6 and 8.

a. According to the definition, the geometric mean is the square

root of 6 x 8.

b. 6 x 8 = 48.

c. 48 = 16 3 4 3u

d. The answer is 4 3

Ex [11] Find the geometric mean of 64, 27, and 8.

a. In this example, we are going to have to find the cube root of

the product of these numbers.

b. However, each number is a perfect cube. So take the cube root

of each individual number and multiply them together.

c. 3 64 4 ; 3 27 3 ; 3 8 2 . So 4 x 3 x 2 = 24.

d. The answer is 24.

Palindrome - A palindrome is a number (or a word) that is the same

frontward and backward.

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Quicker Maths - KopyKitab · Ex [3] 2.5 pounds = _____ oz. a. There are 16 oz in a pound so multiply 2.5 by 16 or 40. b. The answer is 40. C. If you need more conversions, most can