More Fun than a Root Canal: Mathematics Practice Questions
In This Chapter
Making your brain earn its keep with a handful of practice questions
Testing your patience with the dreaded word problem
You’ve had so much fun reviewing algebra, geometry, and trigonometry that you simply can’t wait to jump right in and practice what you know, right? Well, we don’t want you to have to wait any longer to strut your stuff. Heres a set of a dozen math practice questions give ‘em your best shot!
Directions: Each of the following questions has five answer choices. Choose the best answer for each question.
1.
(A)
(B)
(C)
(D)
(E)
First, do the operation inside the parentheses. When you multiply like bases, you add the exponents: . When you have a power outside the parentheses, you multiply the exponents: . Finally, when you divide by like bases, you subtract the exponents: . The correct answer is Choice (B).
If you picked Choice (D), you fell for a trap answer. If you said and , you may have divided by and gotten . If you chose Choice (A), you fell for another trap. You may have reasoned that . Because 12 squared is 144, you may have thought that and that .
All these trap answers are intentional, put there to test whether you know how to perform operations with exponents. If you’re still confused about how to multiply and divide like bases, turn to Chapter 8.
2.The ratio of knives to forks to spoons in a silverware drawer is 3:4:5. Which of the following could be the total number of knives, forks, and spoons in the drawer?
(F)60
(G)62
(H)64
(J)65
(K)66
The total number of utensils must be a multiple of the sum of the numbers of the ratios. In other words, add . The total must be a multiple of 12. Only one answer choice 60 divides evenly by 12, so you know the correct answer is Choice (F).
If you’re confused about ratios (supposedly one of the easiest portions of the exam), check out Chapter 8.
3.An usher passes out 60 percent of his programs before the intermission and 40 percent of the remainder after the intermission. At the end of the evening, what percent of the original number of programs does the usher have left?
(A)60
(B)40
(C)24
(D)16
(E)0
Whenever you have a percentage problem, plug in 100 for the original total. Assume that the usher begins with 100 programs. If he passes out 60 percent of them, he has passed out 60, leaving him with 40. Now comes the tricky part. After the intermission, the usher passes out 40 percent of the remaining programs: 40 percent of 40 is 16 () and . So the correct answer is Choice (C).
Did you fall for the trap answer in Choice (E)? If you thought the usher first passed out 60 programs and then passed out the remaining 40, you believed that he had no programs left at the end of the evening. The word remainder is the key to this problem. The usher didnt pass out 40 percent of his original total, but 40 percent of the remaining programs.
If you chose Choice (D), you made a careless mistake. The number 16 represents the percentage of programs the usher passed out after the intermission. The question asks for the percent of programs the usher had left. We suggest that you circle the portion of the question that tells you what you’re looking for. When you doublecheck your work, review this circled portion first.
4.A salesman makes a commission of $1.50 per shirt sold and $2.50 per pair of pants sold. In one pay period, he sold 10 more shirts than pairs of pants. If his total commission for the pay period was $215, what was the total number of shirts and pairs of pants he sold?
(F)40
(G)50
(H)60
(J)110
(K)150
Let x be the number of pairs of pants the salesman sold. The number of shirts is (because the problem tells you that the salesman sold 10 more shirts than pairs of pants). Set up the following equation:
Now just follow these steps to solve for x:
Multiply:
Combine like terms:
Isolate the x on one side:
Subtract:
Divide:, or
If you answered with Choice (G), you fell for the trap answer (after all that hard work)! Remember to go back and reread what the question is asking for. In this case, it wants to know the total number of pants and shirts sold. So you’re not done working yet. If x (which equals 50) is the number of pairs of pants, then (which is 60) is the number of shirts sold. (Note that 60 is a trap answer as well.) Combine to get the right answer, 110. The correct answer is Choice (J).
5.Kim and Scott work together stuffing envelopes. Kim works twice as fast as Scott. Together they stuff 2,100 envelopes in four hours. How long would Kim working alone take to stuff 175 envelopes?
(A)20 minutes
(B)30 minutes
(C)1 hour
(D)3 hours
(E)6 hours
The ratio of Kims work to Scotts work is 2:1. In other words, she does two out of every three envelopes. Scott does one out of every three envelopes, for a total of 700 envelopes (). Scott stuffs 700 envelopes in four hours, and Kim stuffs 1,400 () in four hours. Divide 1,400 by 4 to find that Kim produces 350 stuffed envelopes per hour. 175 is one half of 350. Therefore, in one half-hour (or 30 minutes), Kim can stuff 175 envelopes. The correct answer is Choice (B).
When you encounter a word problem like this one, don’t start thinking about equations immediately. Talking through the problem may help you more than creating a bunch of equations.
6.If and point O is the center of the circle, what is the shaded area in the figure?
(F)
(G)
(H)
(J)
(K)
A shaded area is the leftover portion of a figure. To find a shaded area, you usually find the total area and the unshaded area and then subtract. In this figure, the shaded area is the total area of rectangle ABCD less the area of half the circle. If the side DC is 6, the radius of the circle is also 6.
The area of a circle is ; therefore, the area of this circle is or . Be careful to remember that you’re working only with a semicircle. The shaded area subtracts only half the area of the circle, so you know you have to subtract . That immediately narrows the answers to Choices (F) and (J).
Next, find the area of the rectangle. (The area of a rectangle equals .) The width of DC is 6. Because the radius of the circle is 6, the diameter of the circle is 12. So BC, the diameter of the circle, is the same as the length of the rectangle. To find the area of the rectangle, simply multiply: . Finally, subtract: . The correct answer is Choice (F).
Shaded area questions should be one of the easiest types of questions to get correct. If you got confused on this problem, flip to Chapter 9.
7.When , what is the value of a?
(A)2
(B)3
(C)4
(D)5
(E)6
First, deal with the parentheses: , which is . Then add like terms: . Finally, solve the equation for a:
The correct answer is Choice (A).
Choice (C) is the trap answer. If you divided 120 by 30 and got 4, you may have picked Choice (C), forgetting that 4 represented , not a.
Of course, you also could simply plug in each answer choice and work backward to solve this problem. Start with the middle value in the answer choices. If , then
The value of Choice (C) is too great, so try Choices (A) and (B), which are smaller numbers.
8.Three times as much as less than 3x is how much in terms of x?
(F)9x
(G)8x
(H)6x
(J)x
(K)
Working backward in this type of problem is usually the easiest way to solve it. Onethird less than 3x is 2x. You can calculate it this way: . Then just multiply by 3: . The correct answer is Choice (H).
9.The following chart shows the weights of junior high school students. What is the sum of the mode and the median weights?
Weight in Pounds
Number of Students
110
4
120
2
130
3
140
2
(A)230 pounds
(B)235 pounds
(C)250 pounds
(D)255 pounds
(E)258 pounds
This question tests vocabulary as much as it tests math. The mode is the most frequently repeated number. In this case, 110 is repeated more often than any other term. The median is the middle term when the numbers are arranged in order. Here you have 110, 110, 110, 110, 120, 120, 130, 130, 130, 140, 140. Of these 11 numbers, the sixth one, 120, is the median. And , so the correct answer is Choice (A).
Don’t confuse median with mean. The mean is the average. You get the mean by adding all the terms and then dividing by the number of terms. If you confused median with mean, you’d really be in a quandary, because the sum of the mean and the mode is 232.73 and that answer isnt an option. If you picked Choice (B), you fell into a different trap. You found that 125 was the median by adding the first and last terms and dividing by 2. Sorry. To find the median, you have to write out all the terms from least to greatest (all four 110s, both 120s, and so on) and then locate the middle term.
10.Points E, D, and A are colinear. The ratio of the area of to is
(F)3:2
(G)3:1
(H)2:1
(J)1:1
(K)1:2
The area of a triangle is . The base of EBD is ED. ED is equal to AD, which is the base of . The bases of the two triangles are equal. The heights are equal, as well. By definition, the height of a triangle is a line from the tallest point perpendicular to the base. If the triangles have the same base and the same height, the ratio of their areas is 1:1. So the correct answer is Choice (J).
11.
Solve for the sum of a + b + c.
(A)15
(B)14
(C)13
(D)12
(E)11
If you’re rushed for time, this problem is a good one to skip and come back to later. Mark a guess on your answer sheet, put a big checkmark next to the questions in your test booklet, and evaluate the question after you’ve finished the last math question. Remember that the ACT doesn’t assess a penalty for wrong answers. Never leave an answer blank. Even a wild guess is worthwhile. However, if you do a few of these practice problems, you’ll be surprised at how quickly you can get them right.
Don’t panic. This problem is much easier than it appears. Start with the righthand column, the ones or units column: a number that ends in 2. You know that the 2 must be a 12 instead of just a 2 because you can’t add a positive number to 5 and get 2, which means . Jot down .
When you carry the 1 to the tens column, you get , which is 8, and . You don’t know a yet . . . or do you? Go to the far-left column (the thousands column). If the answer is ab3 a2, the variable a must equal 1. You can’t add two four-digit numbers and get 20,000-something. The most you can get would be 10,000-something (for example, ). Now you know that a is 1. Jot down .
Go back to the tens column: and (it can’t be 1; it must be 11). Therefore, . Carry the 1 to the hundreds column: and (which is 7) = 13. Yes, this is true—a good check. Carry the 1 to the next column: and is 13, which is what we said ab was in the first place. Therefore, , , , and . The correct answer is Choice (E).
The most common mistake that students make on this type of problem is forgetting to carry the 1 to the next column. Double-check that you have done so.
12. What is the value of ?
(F)14
(G)14.5
(H)15
(J)16
(K)16.5
This problem is a symbolism problem, one that you should think through in words instead of heading for an equation. The symbol indicates that you add the reciprocals of the two numbers. For example, the reciprocal of a is , and the reciprocal of b is . Therefore, add the reciprocals of and . The correct answer is Choice (K).
The has this meaning for this problem only. The meanings of symbols vary from problem to problem; always read the problems carefully.
Making your brain earn its keep with a handful of practice questions
. When you have a power outside the parentheses, you multiply the exponents: 
. Finally, when you divide by like bases, you subtract the exponents: 
. The correct answer is Choice (B).
If you picked Choice (D), you fell for a trap answer. If you said 
and 
, you may have divided 
by 
and gotten 
. If you chose Choice (A), you fell for another trap. You may have reasoned that 
. Because 12 squared is 144, you may have thought that 
and that 
.
. The total must be a multiple of 12. Only one answer choice 60 divides evenly by 12, so you know the correct answer is Choice (F).
) and 
. So the correct answer is Choice (C).
(because the problem tells you that the salesman sold 10 more shirts than pairs of pants). Set up the following equation:
, or 
(which is 60) is the number of shirts sold. (Note that 60 is a trap answer as well.) Combine 
to get the right answer, 110. The correct answer is Choice (J).
). Scott stuffs 700 envelopes in four hours, and Kim stuffs 1,400 (
) in four hours. Divide 1,400 by 4 to find that Kim produces 350 stuffed envelopes per hour. 175 is one half of 350. Therefore, in one half-hour (or 30 minutes), Kim can stuff 175 envelopes. The correct answer is Choice (B).
When you encounter a word problem like this one, don’t start thinking about equations immediately. Talking through the problem may help you more than creating a bunch of equations.
and point O is the center of the circle, what is the shaded area in the figure?
The area of a circle is 
; therefore, the area of this circle is 
or 
. Be careful to remember that you’re working only with a semicircle. The shaded area subtracts only half the area of the circle, so you know you have to subtract 
. That immediately narrows the answers to Choices (F) and (J).
.) The width of DC is 6. Because the radius of the circle is 6, the diameter of the circle is 12. So BC, the diameter of the circle, is the same as the length of the rectangle. To find the area of the rectangle, simply multiply: 
. Finally, subtract: 
. The correct answer is Choice (F).
, what is the value of a?
, which is 
. Then add like terms: 
. Finally, solve the equation for a:
, not a.
, then
less than 3x is how much in terms of x?
. Then just multiply by 3: 
. The correct answer is Choice (H).
, so the correct answer is Choice (A).
to 
is
. The base of EBD is ED. ED is equal to AD, which is the base of 
. The bases of the two triangles are equal. The heights are equal, as well. By definition, the height of a triangle is a line from the tallest point perpendicular to the base. If the triangles have the same base and the same height, the ratio of their areas is 1:1. So the correct answer is Choice (J).
a number that ends in 2. You know that the 2 must be a 12 instead of just a 2 because you can’t add a positive number to 5 and get 2, which means 
. Jot down 
.
, which is 8, and 
. You don’t know a yet . . . or do you? Go to the far-left column (the thousands column). If the answer is ab3 a2, the variable a must equal 1. You can’t add two four-digit numbers and get 20,000-something. The most you can get would be 10,000-something (for example, 
). Now you know that a is 1. Jot down 
.
and 
(it can’t be 1; it must be 11). Therefore, 
. Carry the 1 to the hundreds column: 
and 
(which is 7) = 13. Yes, this is true—a good check. Carry the 1 to the next column: 
and 
is 13, which is what we said ab was in the first place. Therefore, 
, 
, 
, and 
. The correct answer is Choice (E).
What is the value of 
?
indicates that you add the reciprocals of the two numbers. For example, the reciprocal of a is 
, and the reciprocal of b is 
. Therefore, add the reciprocals of 
and 
. 
The correct answer is Choice (K).
has this meaning for this problem only. The meanings of symbols vary from problem to problem; always read the problems carefully.