Math-tanong CEER 2012 - Set 1 solutions

Math-
Math-tanong PRACTICE
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?

(a) 121
(b) 151
(c) 181
(d) 211

math-tanong CEER Supplement set 1

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 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
TIP: For this type of problem, just subtract the rates
before applying the percentage – no need to multiply
first before subtracting.
Those who rented action
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,
75np = 52 i 3 i np
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
75np = 52 i 3 i 5 i 32 = 53 i 33
Hence, the smallest possible value for n + p is
5 + 9 = 14

Question 4
For real numbers a and b, define the operation
a ∗ b = 3b − 2 a
If 2 ∗ 3 = 3 ∗ X , what is the value of X?
?

11
( a) 2 ( c)
3
9
( b) ( d) 4
2


Question 4 Solution
By definition of the operation "∗ ":

2 ∗ 3 = 3 ∗ X ⇒ 3 ( 3 ) − 2 ( 2 ) = 3X − 2 ( 3 )
5 = 3x − 6
3 x = 11
11
x=
3


Question 4 Solution
Of course, you can do substitution. By the definition,
2 ∗ 3 = 3 ( 3 ) − 2 (2 ) = 5 and
3 ∗ X = 3 ( X ) − 2 ( 3 ) = 3X − 6

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
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)

No. of Points Equation :
Equatio n
questions per Total points
answered question 4 x − ( 25 − x ) = 70
Correct x 4 4x 4 x − 25 + x = 70
1(25 − x)
5 x = 95
Wrong 25 − x 1
= 25 − x x = 19


Question 6
In the figure below, 1 2 . What is the
value of x?

(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 and 2

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:

m = 20 ( corresponding angles )
n = 180 − 150 = 30
(alternate interior 150, m°
thenadjacent) n°

∴ x = m + n = 50

Question 7
If 6 − x = 5.43 , what is the value of 6 + x ?

(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 − x = 5.43 ⇒ − x = 5.43 − 6 = −0.57
x = 0.57
∴ 6 + x = 6 + 0.57 = 6.57


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
S + 40
average = = 38 ⇒ S + 40 = 190
5
But the last number should be 30 instead of 40, so
S + 40 − 10 = 190 − 10 ⇒ S + 30 = 180
The average, then, should have been
S + 30 180
= = 36
5 5

Question 9 Solution
“Expert” shortcut solution: Start with
S + 40
original average = = 38 ⇒ S + 40 = 190
5
Then the correct average is
S + 30 ( S + 40 ) − 10 190 − 10 180
= = = = 36
5 5 5 5


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?

3
( a) 20 ( c) 20 −
2
3
( b) ( 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
to CB, and AE acts y y
as a transversal, then 2 x 3
30°
angle AEF is 30 2
degrees. Hence, x x
triangle EAF is also a 2 x 2
30-60-90 triangle.

Hence, we have the following:
1. By the properties of a 30-60-90 triangle:

 6  12 3
EC = 2   = i =4 3
 3 3 3
30°
x 3
=6
2
1
DC = EC = 2 3 x
2 x 2



2. Since BCDF is a rectangle, DC = BF, so
DC = BF = 2 3

3. Since AB = 10, then
AF = 10 − BF 30°

= 10 − 2 3 2 3 2 3



4. Since EAF is a 30-60-90 triangle with angle AEF
= 30 degrees, then

AE = 2AF
(
y = 2 10 − 2 3 )
(
= 2 10 − 2 3 ) 10 − 2 3
30°

= 20 − 4 3 2 3


Expert fast solution! Let x = EF and z = AE Based on the
established facts earlier, you have the figure shown.
x 3 12 3
=6⇒x = i =4 3
2 3 3
x 4 3 z
= =2 3 z
2 2 y =
2 x 3
x 30° =6
y = 10 − = 10 − 2 3 2
2
x x
z
2
(
y = = 10 − 2 3 ⇒ z = 2 10 − 2 3 ) 2 x 2
= 20 − 4 3

Math-tanong CEER 2012 - Set 1 solutions

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