Math-tanong CEER 2012 - Set 1 Solutions

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SET 1 with Solutions

Question 1

What is the smallest three-digit number that

leaves a remainder of 1 when divided by 2,

3, and 5?

set 1math-tanong CEER Supplement

(a) 121

(b) 151

(c) 181

(d) 211

Question 1 Solution

• Determine first the least common multiple

(LCM) of 2, 3, and 5 – which is 30.

• Now, find the smallest three-digit multiple of

the LCM – that’s 120.the LCM – that’s 120.

• The smallest 3-digit number that leaves a

remainder of 1 when divided by 2, 3, and 5 is

the smallest 3-digit number that leaves a

remainder of 1 when divided by 30, the LCM

– that’s 121 ☺


Question 2

The pie chart above shows the distribution of DVD

rentals from Dibidi Doo Bee shop for a single night. If

250 DVDs were rented that night, how many more

action movies were rented than horror movies?


(a) 10

(b) 20

(c) 22

(d) 25

Question 2 Solution

Those who rented action

movies are

TIP: For this type of problem, just subtract the rates

before applying the percentage – no need to multiply

first before subtracting.

movies are

22% — 12% = 10%

more than those who rented

horror movies. Thus, there are

250(10%) = 250(0.1) = 25

more action movies rented than

horror movies.


Question 3

Suppose that n and p are integers greater than 1. If 5n is

a perfect square and 75np is a perfect cube, what is the

smallest value of n + p?


(a) 5 (c) 8

(b) 6 (d) 15

Question 3 Solution

If 5n is a perfect square, the smallest possible value of n

would be 5. Now,275 5 3np np= i i

If 75np is a perfect cube and n = 5 is the smallest


If 75np is a perfect cube and n = 5 is the smallest

possible value of n, the smallest possible value for p is

32 = 9 since 2 2 3 375 5 3 5 3 5 3np = =i i i i

Hence, the smallest possible value for n + p is

5 + 9 = 14

Question 4

For real numbers a and b, define the operation

3 2a b b a∗ = −

If , what is the value of X? 2 3 3 X∗ = ∗


If , what is the value of ? 2 3 3 X∗ = ∗

( ) ( )

( ) ( )

11a 2 c

3

9b d 4

2

Question 4 Solution

By definition of the operation :

( ) ( ) ( )2 3 3 3 3 2 2 3 2 3

5 3 6

X X

x

∗ = ∗ ⇒ − = −

= −

" "∗


5 3 6

3 11

11

3

x

x

x

= −

=

=

Question 4 Solution

Of course, you can do substitution. By the definition,

( ) ( )

( ) ( )

2 3 3 3 2 2 5

3 3 2 3 3 6

and

X X X

∗ = − =

∗ = − = −


( ) ( )3 3 2 3 3 6X X X∗ = − = −

Q: Which of the choices will give a value of 5 when

substituted to X in 3X − 6? Choice (c).

Question 5

A test is composed of 25 questions. The score of an

examinee is obtained by giving him 4 points for each

correct answer and deducting 1 point for each wrong

answer. If an examinee answered all the questions and

obtained a score of 70, how many questions did he obtained a score of 70, how many questions did he

answer correctly?


(a) 17 (c) 19

(b) 18 (d) 20

Question 5 Solution

Let x = the number of correct answers

25 – x = the number of wrong answers

(since there are 25 questions in all)

:Equation


No. of questions answered

Points per

questionTotal points

Correct x 4 4x

Wrong 25 − x 11(25 − x) = 25 − x

( )4 25 70

4 25 70

5 95

9

:

1

Equatio

x x

x x

x

x

n

− − =

− + =

=

=

Question 6

In the figure below, . What is the

value of x?1 2� � �

(a) 40


(a) 40

(b) 50

(c) 60

(d) 70

`

Question 6 Solution

Draw an extra line passing through the vertex of

angle x and parallel to 1 2 and � �


m°

n°

Question 6 Solution

If m and n are the angles formed as in the figure, then,

using the ideas of angles formed by a transversal:

( )20 corresponding anglesm =


m°

n°

180 150 30

(alternate interior 150,

thenadjacent)

n = − =

50x m n∴ = + =

Question 7

If , what is the value of ? 6 5.43x− = 6 x+

(a) 6.43


(a) 6.43

(b) 6.57

(c) 17.43

(d) 17.51

Question 7 Solution

Warning: DO NOT ATTEMPT to solve

for x! Sayang ang effort!


6 5.43 5.43 6 0.57

0.57

6 6 0.57 6.57

x x

x

x

− = ⇒− = − = −

=

∴ + = + =

Question 9

Erin calculated the average of 5 numbers to be 38. Then

she found out that she had made an error and had

written 40 for one of the numbers when she should have

written 30. What is the average of the correct 5 numbers?


(a) 32

(b) 34

(c) 36

(d) 38

Question 9 Solution

Let S be the sum of the first 4 numbers that Erin has.

If the average of 5 numbers is 38, including the

erroneous 40, then

4038 40 190

5

Saverage S

+= = ⇒ + =


38 40 1905

average S= = ⇒ + =

But the last number should be 30 instead of 40, so

40 10 190 10 30 180S S+ − = − ⇒ + =

The average, then, should have been

30 18036

5 5

S+= =

Question 9 Solution

“Expert” shortcut solution: Start with

4038 40 190

5

Soriginal average S

+= = ⇒ + =


5

Then the correct average is

( )40 1030 190 10 18036

5 5 5 5

SS + −+ −= = = =

Question 10

In the diagram, FDCB is a rectangle. Segment ED is 6

units long, segment AB is 10 units long, and the measure

of angle ECD is 60°.What is the length of segment AE?


( ) ( )

( ) ( )

3a 20 c 20

2

3b d 20 4 3

2

−

−

Question 10 Solution

Fast facts about the figure:

1. Triangle CDE is a 30-60-90 triangle with angle DEC =

30 degrees.

2. Since DF is parallel


2. Since DF is parallel

to CB, and AE acts

as a transversal, then

angle AEF is 30

degrees. Hence,

triangle EAF is also a

30-60-90 triangle.

30°

x2

x

3

2

x

y

2

y

2

x


Hence, we have the following:

1. By the properties of a 30-60-90 triangle:


6 12 32 4 3

3 3 3

12 3

2

EC

DC EC

= = =

= =

i

30°

x2

x

36

2

x=


2. Since BCDF is a rectangle, DC = BF, so

2 3DC BF= =


3. Since AB = 10, then

10

10 2 3

AF BF= −

= −

30°

2 32 3


4. Since EAF is a 30-60-90 triangle with angle AEF

= 30 degrees, then

2AE AF=


( )

2

2 10 2 3

20 4 3

AE AF=

= −

= −

30°

10 2 3−

2 3

( )2 10 2 3y = −


Expert fast solution! Let x = EF and z = AE Based on the

established facts earlier, you have the figure shown.

3 12 36 4 3

2 3 3

xx= ⇒ = =i


30°

x2

x

36

2

x=

z

2

zy =

2

x

( )

4 32 3

2 2

10 10 2 32

10 2 3 2 10 2 32

20 4 3

x

xy

zy z

= =

= − = −

= = − ⇒ = −

= −

END OF PRACTICE SET

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Math-tanong CEER 2012 - Set 1 Solutions