Question 1

For which value of x is the following inequality NOT true? A -1
B -1/4
C 1/4
D 1/2
E 4

Question 2

If 3v + 2(1 – v) = 7, what is v?

A 1
B 7/5
C 3
D 4
E 5

Question 3

Bob builds 3 barrels per hour. George builds 2 barrels per hour. Working simultaneously, how many hours will it take them to build 15 barrels?

A 1 1/2
B 2
C 2 1/2
D 3
E 3 1/2

Question 4

z + 13 = 0
y – 7 = 0

What is the value of z – y?

A -20
B -6
C 0
D 6
E 20

Question 5

If 2x+4=16, then what is the value of x+2?

A 3
B 4
C 8
D 10
E 12

Question 6

If apples cost 50 cents for three, and oranges cost 70 cents for two, how much does it cost to buy six apples and six oranges?

A \$2.40
B \$2.70
C \$3.10
D \$3.40
E \$3.60

Question 7 What is the value of x?

A (2 – 6)/3
B 2 – 6
C 2 – 4
D (6 – 4)/3
E (3 – 12)/2

Question 8

If u = 17 and 2uv = 68 then v =

A 2
B 3
C 4
D 5
E 6

Question 9

(170/y) < x2 < (300/y)

When y = 2, what is one possible integer value of x?

A 12
B 9
C 11
D 13
E 10

Question 10

For what value of x is an integer?

A 3
B 4
C 5
D 7
E 9

Question 11

If , then x =

A 3
B 6
C 9
D 27
E 81

Question 12

(3 + a + 5)2 > 144

If a is an integer, what is the minimum value that makes the above statement true?

A 0
B 3
C 4
D 5
E 13

Question 13

Soldier X can polish 4 pairs of boots per hour, while Soldier Y can polish 3 per hour. If Soldier Y starts 2 hours before Soldier X and both soldiers polish without rest, how many hours will it take Soldier X to catch up so that they have each polished the same number of pairs of boots?

A 1
B 2
C 4
D 5
E 6

Question 14

If it takes 12 people 3 days to build 72 meters of fence, how many meters of fence can 15 people build in one day?

A 2
B 12
C 20
D 24
E 30

Question 15

st + 1 = 1
t = 3

What is the value of s?

A -1
B -1/3
C 0
D 1/3
E 1

Question 16

A square foot of carpet costs D dollars. What is the cost of covering a room that has dimensions 9 feet by 8 feet?

A 9D
B 17D2
C 9D + 8D
D 72D2
E 72D

Question 17

A student decides to sell lemonade at her school. She knows she will sell 90 glasses of lemonade each day. Each glass will cost her 30 cents to make. What is the minimum price she should charge per glass if she hopes to make \$45 profit each day?

A 35 cents
B 45 cents
C 50 cents
D 72 cents
E 80 cents

Question 18

If a book has 250 words per page, Sammy can read 3 pages in 5 minutes. If the book has 300 words per page, Sammy can read 5 pages in 10 minutes. How many words per second can Sammy read?

A 2
B 2.5
C 3
D 3.5
E 5

Question 19

A softball team played 30 games, and lost 5 times as many games as they won. If there were no ties, how many games did they lose?

A 25
B 20
C 15
D 10
E 5

Question 20

Suppose that flushing a toilet uses 4 gallons of water, and taking a shower uses 2 gallons per minute. How much water is used per day in a house where the toilet is flushed 9 times and there are 3 six minute showers?

A 76
B 72
C 54
D 48
E 36

1. E
The general strategy of plugging in numbers here will get you to the right answer. You can speed things up by noticing that there is an x2 in the denominator of one of the left hand terms. You should notice that is x is a fraction, this term will become quite large, and the inequality is likely to hold. You can quickly check that only x = 4 violates the inequality.

2. E
Simplifying the left hand side of the equation we get:
3v + 2 – 2v = v + 2 = 7. And so v = 5.

3. D
Bob can build 3 an hour; George can build 2, so together they can build 5 in an hour. To build 15 barrels, they need 15/5=3 hours.

4. A
If you subtract the two equations from each other, you get:

z – y + 13 -(-7) = 0
which yields
z – y = -20

Alternatively, you can solve for z = -13 and y = 7 right away.

5. C
Solve this one with a quick trick: we know 2x+4=16. Divide both sides of this equation by 2. We obtain 1x+2=8. That’s the answer we need. Sometimes the math problems are set up so that you can use a simple trick like this one.

6. C
Three apples were 50 cents, so six are \$1.00. Two oranges are 70 cents, so six are \$2.10. Now add the two together to get \$3.10. (Note: you can add apples and oranges when you talk about them both in terms of price.)

7. C
Divide the top and bottom of the left by 3 to give (x + 4)/2. Next, multiply each side of the equation by 2 to get: x + 4 = 2 . Finally, subtract 4 from each side to get x = 2 – 4.

8. A
Plugging in for u in the second equation, we get 34v = 68, which means that v = 2.

9. A, C, E
Substituting 2 for y in the expression leaves us with the inequality:
(85) < x2 < (150)
So x can take on any of the values 10, 11 and 12. Any other value would make x2 too small or too large to satisfy the inequality.

10. D
If you spot the trick you can avoid tedious math. Since 7 will be added to 6x, when x = 7, the numerator is guaranteed to be divisible by 7. You can also see this by rewriting the expression as 6 + 7/x.

11. E
We need the number whose square root is 9. Well, 9 times 9 is 81, so that’s the number we’re looking for.

12. D
The first step is to simplify the expression to (a + 8)2 > 144. Now 144 is 12 squared, so a + 8 has to be greater than 12. The minimum value for a is 5. (If a = 4, the inequality does not hold.)

13. E
This one is quite straightforward. Just notice that by starting two hours before Soldier X, Soldier Y managed to polish 6 pairs of boots. Now, as Soldier X polishes one more pair per hour, it will take her 6 hours to catch up.

You can also write this in equation form. Let t be the number of hours that Soldier X and Soldier Y work simultaneously. Then we have:
4t = 3t + 6 which implies that t = 6.

14. E
Start by figuring out how many meters of fence the 12 people can build in one day. If they can do 72 in 3 days, that means they can build 24 meters in one day. You can then figure out that one person builds 2 meters of fence per day. Or, you could quickly notice that 15 people are 25% more than 12 people, and they should be able to build 25% more fence. The final answer is, indeed, 30 meters.

15. C
The first equation shows that st = 0, so when the second equation tells us that t is not zero, the only possibility is that s = 0.

16. E
The room has 72 square feet, and each square foot costs D dollars. The total cost is 72D.

17. E
The quickest way to the answer is to see that if she sells 90 glasses and makes \$45 she is making 50 cents per glass. Therefore she should charge 30 + 50 = 80 cents.

Alternatively, you could solve the following equation:

45 = 90*(x – .30) = 90x – 90*.3 = 90x – 27

The amount of money she makes (\$45) is 90 times the difference between the price of the lemonade and the cost of the lemonade (i.e. 90 times her profit per glass). It follows that 90x = 72 and that x = 0.80.

18. B
This is actually easier than it looks. There is only one unknown quantity — the number of words per second that Sammy can read. Therefore, there is too much information in the question. Only one of the equations would have been sufficient.

Words per second is found by dividing the total number of words by the total number of seconds. Taking the first sentence, we see that Sammy can read 250 * 3 = 750 words in 300 seconds. (He reads 3 pages with 250 words on each page in 5 minutes.) So we just divide 750 by 300 to get 2.5 words per second.

Did you know that you’re also automatically enrolled in our vocabulary builder tutorial? Why not visit the vocabulary builder sometime and check it out? This link will open in a new window, so you can continue what you’re doing now uninterrupted.

19. A
You have two unknowns here, the number of losses (L) and the number of wins (W). So you need two distinct pieces of information to solve for L and W. First, you know that L + W = 30. Then you also know that L = 5W. Substituting in the first equation gives you 5W + W = 30, which implies that W = 5. Therefore, the team lost 25 games (L = 25).

20. B
The general equation is 4x + 2ty where x is the number of toilet flushes, y is the number of showers and t is the average length of the showers. In this case we have 4*9 + 2*6*3 = 72.