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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 ☺
 math-tanong CEER Supplement                  set 1
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


 math-tanong CEER Supplement                          set 1
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.
 math-tanong CEER Supplement                             set 1
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



 math-tanong CEER Supplement                           set 1
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
  math-tanong CEER Supplement                                 set 1
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

 math-tanong CEER Supplement                     set 1
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


 math-tanong CEER Supplement                         set 1
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).



  math-tanong CEER Supplement                     set 1
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


 math-tanong CEER Supplement                            set 1
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


    math-tanong CEER Supplement                                          set 1
Question 6
In the figure below,           1   2   . What is the
value of x?

(a)     40
(b)     50
(c)     60
(d)     70


 math-tanong CEER Supplement                           set 1
Question 6 Solution
Draw an extra line passing through the vertex of
angle x and parallel to 1 and 2



                               m°
                              n°



math-tanong CEER Supplement                   set 1
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
  math-tanong CEER Supplement                              set 1
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



 math-tanong CEER Supplement                     set 1
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

 math-tanong CEER Supplement                         set 1
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

  math-tanong CEER Supplement                         set 1
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
  math-tanong CEER Supplement                            set 1
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


 math-tanong CEER Supplement                       set 1
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

 math-tanong CEER Supplement                         set 1
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.
 math-tanong CEER Supplement                        set 1
Question 10 Solution
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


  math-tanong CEER Supplement                       set 1
Question 10 Solution

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



 math-tanong CEER Supplement                 set 1
Question 10 Solution

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




 math-tanong CEER Supplement                                       set 1
Question 10 Solution
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 Supplement                               set 1
END OF PRACTICE SET

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Math-tanong CEER 2012 - Set 1 solutions

  • 1.
  • 2. Math- Math-tanong PRACTICE SET 1 with Solutions
  • 3. 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
  • 4. 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 ☺ math-tanong CEER Supplement set 1
  • 5. 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 math-tanong CEER Supplement set 1
  • 6. 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. math-tanong CEER Supplement set 1
  • 7. 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 math-tanong CEER Supplement set 1
  • 8. 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 math-tanong CEER Supplement set 1
  • 9. 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 math-tanong CEER Supplement set 1
  • 10. 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 math-tanong CEER Supplement set 1
  • 11. 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). math-tanong CEER Supplement set 1
  • 12. 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 math-tanong CEER Supplement set 1
  • 13. 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 math-tanong CEER Supplement set 1
  • 14. Question 6 In the figure below, 1 2 . What is the value of x? (a) 40 (b) 50 (c) 60 (d) 70 math-tanong CEER Supplement set 1
  • 15. Question 6 Solution Draw an extra line passing through the vertex of angle x and parallel to 1 and 2 m° n° math-tanong CEER Supplement set 1
  • 16. 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 math-tanong CEER Supplement set 1
  • 17. 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 math-tanong CEER Supplement set 1
  • 18. 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 math-tanong CEER Supplement set 1
  • 19. 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 math-tanong CEER Supplement set 1
  • 20. 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 math-tanong CEER Supplement set 1
  • 21. 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 math-tanong CEER Supplement set 1
  • 22. 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 math-tanong CEER Supplement set 1
  • 23. 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. math-tanong CEER Supplement set 1
  • 24. Question 10 Solution 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 math-tanong CEER Supplement set 1
  • 25. Question 10 Solution 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 math-tanong CEER Supplement set 1
  • 26. Question 10 Solution 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 math-tanong CEER Supplement set 1
  • 27. Question 10 Solution 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 Supplement set 1