MATH LEARNING QUIZ:
1. Oil is pumped into a non-empty tank at a changing rate. The volume of oil in the tank doubles every
minute and the tank is filled in 10 minutes. How many minutes did it take for the tank to be half full?
A. 2 B. 5 C. 7 D. 8 E. 9
Answer: (e)
Solution: At the 10 minute mark, the tank has twice as much oil in it as it did at the 9 minute mark. So the
tank became half-full at the 9 minute mark.
Area: Algebra
2. Alice and Bill are walking in opposite directions along the same route between A and B. Alice is going
from A to B, and Bill from B to A. They start at the same time. They pass each other 3 hours later. Alice
arrives at B 2.5 hours before Bill arrives at A. How many hours does it take for Bill to go from B to A?
A. 6 B. 6.5 C. 7 d. 7.5 E. 8.5
Answer: (d)
Area: Algebra
Solution: Suppose that it takes Alice x hours to go from A to B. Then, it takes Bill x +2.5 hours to go
from B to A. Assuming that Alice and Bill maintain constant speed, we get that Alice’s speed is d=x and
Bill’s speed is d=(x +2.5), where d is the distance from A to B. Since Alice and Bill met 3 hours after they
started walking,
Dividing by d and solving for x we get x =5, so the answer is 5+2:5=7:5 hours.
Area: Algebra
3. Two armies are advancing towards each other, each one at 1 mph. A messenger leaves the first army
when the two armies are 10 miles apart and runs towards the second at 9 mph. Upon reaching the second
army, he immediately turns around and runs towards the first army at 9 mph. How many miles apart are
the two armies when the messenger gets back to the first army?
A. 5.6 b. 5.8 C. 6 D. 6.2 E. 6.4
Answer: (e)
Solution: The messenger reaches the second army in one hour (the messenger and the second army
advance at each other at a combined speed of 10 mph). At that time, the two armies are 8 miles apart. It
takes 0.8 hours for the messenger to get back to the first army. At that time the armies are 8 - 0:8 -0.8=6.4
miles apart.
Area: Algebra
4. You play on a game show where a prize has been randomly put into one of five boxes, labeled A, B, C,
D, and E, with each box equally likely to contain the prize. The boxes are closed when you first see them,
, A. 1/5 B. 1/3 C. 2/5 D. 3/5 E. 2/3
Answer: (c)
Solution: The probability of the prize being in box A is equal to that of it being in any other box; hence
this probability is 1=5. The remaining 4=5 is split between Box D and Box B; hence the probability that
Box D contains the prize is 2=5.
Area: Statistics
5. Six mountain climbers divide themselves into three teams for the final assault on a peak. One team has
3 climbers; the others have 1 and 2 climbers. All manners of different team deployments are considered,
which team goes first, second, and third, and two deployments are considered different if the composition
of any of the teams is different. (Disregard deployments within each team.) What is the total number of
possible team deployments?
A. 60 B. 180 c. 360 D. 720 E. none of the above
Answer: (c)
Solution: There are six ways to choose which team goes first, which goes second and which 6 goes third.
Also, there are = 20 ways to select three climbers out of the six for the three man team, and 3 ways to pick
a climber out of the remaining three for the one man team. So, the total number of ways is 6 .20 .3 = 360
ways.
Area: Statistics
6. How many positive integers less than or equal to 2013 are divisible by at least one of 3, 11, and 61?
A. 3 B. 813 C. 1006 D. 1198 E. 2013
Answer: (b)
Solution: There are 2013/3 = 671 multiples of 3 from 1 to 2013, 2013/11 = 183 multiples of 11, and
2013/61 = 33 multiples of 61. To avoid double counting we subtract the 61 multiples of 3 .11, the 11
multiples of 3 .61, and the 3 multiples of 61 .11. We have to add 1 for 2013 which was subtracted one too
many times. Thus, the answer is 671 + 183 + 33 – 61 – 11 – 3 + 1 = 813.
Area: Algebra
7. How many two digit prime numbers are there in which both digits are prime numbers? (For example,
23 is one of these numbers but 31 is not, since 1 is not a prime number.)
A. 3 B. 4 C. 5 D. 8 E. 15
Answer: (b)
Solution: The second digit can only be 3 or 7, so the choice quickly narrows down to 23, 27,33, 37, 53,
57, 73, and 77. Of these, 27, 33, and 57 are divisible by 3, and 77 by 7, leaving 23, 37, 53, and 73. It is
easy to see that none of these is divisible by 2, 3, 5, or 7, and there is no need to look at greater prime
divisors since √77 < 11.
8. You own thirteen pairs of socks, all different, and all of the socks are individually jumbled in a drawer.
One morning you rummage through the drawer and continue to pull out socks until you have a matching
pair. How many socks must you pull out to guarantee having a matching pair?
A. 3 B. 12 C. 13 D. 14 E. 25
, Solution: You might be unlucky and have the first thirteen socks all different, but then the 14 thhas to
match one of them.
9. A jeweler has a 20 gram ring that is 60% gold and 40% silver. He wants to melt it down andadd enough
gold to make it 80% gold. How many grams of gold should be added?
A. 4 grams B. 8 grams C. 12 grams D. 16 grams E. 20 grams
Answer: (e)
Solution:
10. Consider the following game. A referee has cards labeled A, B, C, and D, and places them face down
in some order. You point to each card in turn, and guess what letter is written on the bottom. You guess
each of A, B, C, and D exactly once (otherwise there is no chance ofgetting them all right!). You play this
game once, and then the referee tells you that you guessed exactly n of the letters correctly. Which value
of n is not a possible value of n?
A. 0 B. 1 C. 2 D. 3 E. 4
Answer: (d)
Solution: If the first three are correct, then by process of elimination the fourth has to be correct also. The
same reasoning holds no matter when the three correct answers occur.
11. Consider the following game. A referee has cards labeled A, B, C, and D, and places them face down
in some order. You point to each card in turn, and guess what letter is written on the bottom. You guess
each of A, B, C, and D exactly once (otherwise there is no chance of getting them all right!).You play this
game once, and then the referee tells you that you guessed exactly n of the letters correctly. Which value
of n is not a possible value of n?
A. 0 B. 1 C. 2 D. 3 E. 4
Answer: (d)
Solution: If the first three are correct, then by process of elimination the fourth has to be correct also. The
same reasoning holds no matter when the three correct answers occur.
12. On a test the passing students had an average of 83, while the failing students had an average of 55. If
the overall class average was 76, what percent of the class passed?
A. 44% B. 66% C. 68% D. 72% E. 75%
Answer: (e)
Solution: Let p = proportion that passed. Then 83 p + 55(1 - p) = 76, so p = 21/ 28 = .75
1. Oil is pumped into a non-empty tank at a changing rate. The volume of oil in the tank doubles every
minute and the tank is filled in 10 minutes. How many minutes did it take for the tank to be half full?
A. 2 B. 5 C. 7 D. 8 E. 9
Answer: (e)
Solution: At the 10 minute mark, the tank has twice as much oil in it as it did at the 9 minute mark. So the
tank became half-full at the 9 minute mark.
Area: Algebra
2. Alice and Bill are walking in opposite directions along the same route between A and B. Alice is going
from A to B, and Bill from B to A. They start at the same time. They pass each other 3 hours later. Alice
arrives at B 2.5 hours before Bill arrives at A. How many hours does it take for Bill to go from B to A?
A. 6 B. 6.5 C. 7 d. 7.5 E. 8.5
Answer: (d)
Area: Algebra
Solution: Suppose that it takes Alice x hours to go from A to B. Then, it takes Bill x +2.5 hours to go
from B to A. Assuming that Alice and Bill maintain constant speed, we get that Alice’s speed is d=x and
Bill’s speed is d=(x +2.5), where d is the distance from A to B. Since Alice and Bill met 3 hours after they
started walking,
Dividing by d and solving for x we get x =5, so the answer is 5+2:5=7:5 hours.
Area: Algebra
3. Two armies are advancing towards each other, each one at 1 mph. A messenger leaves the first army
when the two armies are 10 miles apart and runs towards the second at 9 mph. Upon reaching the second
army, he immediately turns around and runs towards the first army at 9 mph. How many miles apart are
the two armies when the messenger gets back to the first army?
A. 5.6 b. 5.8 C. 6 D. 6.2 E. 6.4
Answer: (e)
Solution: The messenger reaches the second army in one hour (the messenger and the second army
advance at each other at a combined speed of 10 mph). At that time, the two armies are 8 miles apart. It
takes 0.8 hours for the messenger to get back to the first army. At that time the armies are 8 - 0:8 -0.8=6.4
miles apart.
Area: Algebra
4. You play on a game show where a prize has been randomly put into one of five boxes, labeled A, B, C,
D, and E, with each box equally likely to contain the prize. The boxes are closed when you first see them,
, A. 1/5 B. 1/3 C. 2/5 D. 3/5 E. 2/3
Answer: (c)
Solution: The probability of the prize being in box A is equal to that of it being in any other box; hence
this probability is 1=5. The remaining 4=5 is split between Box D and Box B; hence the probability that
Box D contains the prize is 2=5.
Area: Statistics
5. Six mountain climbers divide themselves into three teams for the final assault on a peak. One team has
3 climbers; the others have 1 and 2 climbers. All manners of different team deployments are considered,
which team goes first, second, and third, and two deployments are considered different if the composition
of any of the teams is different. (Disregard deployments within each team.) What is the total number of
possible team deployments?
A. 60 B. 180 c. 360 D. 720 E. none of the above
Answer: (c)
Solution: There are six ways to choose which team goes first, which goes second and which 6 goes third.
Also, there are = 20 ways to select three climbers out of the six for the three man team, and 3 ways to pick
a climber out of the remaining three for the one man team. So, the total number of ways is 6 .20 .3 = 360
ways.
Area: Statistics
6. How many positive integers less than or equal to 2013 are divisible by at least one of 3, 11, and 61?
A. 3 B. 813 C. 1006 D. 1198 E. 2013
Answer: (b)
Solution: There are 2013/3 = 671 multiples of 3 from 1 to 2013, 2013/11 = 183 multiples of 11, and
2013/61 = 33 multiples of 61. To avoid double counting we subtract the 61 multiples of 3 .11, the 11
multiples of 3 .61, and the 3 multiples of 61 .11. We have to add 1 for 2013 which was subtracted one too
many times. Thus, the answer is 671 + 183 + 33 – 61 – 11 – 3 + 1 = 813.
Area: Algebra
7. How many two digit prime numbers are there in which both digits are prime numbers? (For example,
23 is one of these numbers but 31 is not, since 1 is not a prime number.)
A. 3 B. 4 C. 5 D. 8 E. 15
Answer: (b)
Solution: The second digit can only be 3 or 7, so the choice quickly narrows down to 23, 27,33, 37, 53,
57, 73, and 77. Of these, 27, 33, and 57 are divisible by 3, and 77 by 7, leaving 23, 37, 53, and 73. It is
easy to see that none of these is divisible by 2, 3, 5, or 7, and there is no need to look at greater prime
divisors since √77 < 11.
8. You own thirteen pairs of socks, all different, and all of the socks are individually jumbled in a drawer.
One morning you rummage through the drawer and continue to pull out socks until you have a matching
pair. How many socks must you pull out to guarantee having a matching pair?
A. 3 B. 12 C. 13 D. 14 E. 25
, Solution: You might be unlucky and have the first thirteen socks all different, but then the 14 thhas to
match one of them.
9. A jeweler has a 20 gram ring that is 60% gold and 40% silver. He wants to melt it down andadd enough
gold to make it 80% gold. How many grams of gold should be added?
A. 4 grams B. 8 grams C. 12 grams D. 16 grams E. 20 grams
Answer: (e)
Solution:
10. Consider the following game. A referee has cards labeled A, B, C, and D, and places them face down
in some order. You point to each card in turn, and guess what letter is written on the bottom. You guess
each of A, B, C, and D exactly once (otherwise there is no chance ofgetting them all right!). You play this
game once, and then the referee tells you that you guessed exactly n of the letters correctly. Which value
of n is not a possible value of n?
A. 0 B. 1 C. 2 D. 3 E. 4
Answer: (d)
Solution: If the first three are correct, then by process of elimination the fourth has to be correct also. The
same reasoning holds no matter when the three correct answers occur.
11. Consider the following game. A referee has cards labeled A, B, C, and D, and places them face down
in some order. You point to each card in turn, and guess what letter is written on the bottom. You guess
each of A, B, C, and D exactly once (otherwise there is no chance of getting them all right!).You play this
game once, and then the referee tells you that you guessed exactly n of the letters correctly. Which value
of n is not a possible value of n?
A. 0 B. 1 C. 2 D. 3 E. 4
Answer: (d)
Solution: If the first three are correct, then by process of elimination the fourth has to be correct also. The
same reasoning holds no matter when the three correct answers occur.
12. On a test the passing students had an average of 83, while the failing students had an average of 55. If
the overall class average was 76, what percent of the class passed?
A. 44% B. 66% C. 68% D. 72% E. 75%
Answer: (e)
Solution: Let p = proportion that passed. Then 83 p + 55(1 - p) = 76, so p = 21/ 28 = .75
