IN THIS CHAPTER

Honing your GMAT math skills by working through practice questions

Taking a look at the answer explanations to understand what you did wrong — and right

Here’s a chance to test your GMAT math skills before you embark on the real adventure of taking the test. This chapter contains only the types of math questions you’ll see on the GMAT, so it’s kind of like a mini practice test. To get a better idea of the time restrictions you’ll face on test day, try to complete the questions in the following section in about 48 minutes. If you want to avoid the time pressure for now, feel free to just focus on answering the questions. You’ll have the opportunity to time yourself again when you take the full-length practice tests included with this book.

Read through all the answer explanations (even the ones for the questions you answered correctly), because you want to make sure you know why you got the answer you did and because you may see something in the explanations that can help you with other questions.

Tackling GMAT Math Practice Questions

Here are 24 practice questions for the GMAT math section. Grab your pencil, set your timer for 48 minutes, and get started. (Try not to peek at the answers until you’ve come up with your own.)

Checking Out the Answer Explanations

1. A. The GMAT usually starts with a question of medium difficulty, and this one is in that range. If the product of two factors equals 0, then at least one of the factors must be 0 (because anything times 0 equals 0). Therefore, one of the factors in this equation must equal 0. You know it isn’t the second one, because y doesn’t equal 5, and y would have to equal 5 for the second term to result in 0.

   Therefore, you need to create an equation that sets the first factor equal to 0 and then solve for y. Here’s what you get for the first factor:

   

   Cross-multiply (because ) and solve:

   
2. E. If Esperanza will be 35 years old in 6 years, she is 29 right now (). Therefore, to determine how old she was x years ago, simply subtract x from her current age of 29: 29 – x.

3. D. This problem is simple when you recognize that because the two fractions have a common denominator, is the same thing as .

   Statement (1) says that , and because , must also equal 6. So you know that Statement (1) is sufficient to answer the question and that the answer must be either Choice (A) or Choice (D). To figure out which it is, consider Statement (2). If it’s sufficient, the answer is Choice (D). If not, the answer is Choice (A).

   Because , and Statement (2) tells you the value of x + y, you can substitute 18 for x + y in the expression and solve for a known value (). So Statement (2) also provides sufficient information to answer the question.

4. C. This is a percent decrease question. You can apply a formula to solve it, but a faster and easier method is to apply actual numbers to the circumstances. To simplify your life, use a nice, round figure like $100.

   If the couch originally cost $100 but was discounted by 20 percent, you’d multiply $100 by 20 percent (0.20) and subtract that from $100 to find the price after the first discount (, and ). After the first round of discounts, the couch cost $80.

   However, the couch was discounted an additional 20 percent. Now, you have to repeat the process, this time using $80 as the original price (, and ). After both discounts, the couch cost $64.

   But you’re not finished yet. You need to calculate the total discount. The couch originally cost $100 and later cost $64. The discount, in dollars, is 100 – 64, which is $36. To find the percentage of the full discount, simply divide $36 by the original price of $100 ( or 36 percent).

5. E. Evaluate Statement (1). Knowing that x is even doesn’t help you much. Three numbers in the set are even: 44, 52, and 58. So Statement (1) doesn’t allow you to narrow down the value of x to one number. The answer can’t be Choice (A) or Choice (D).

   Consider Statement (2). Two numbers in the set are multiples of 4: 44 and 52. So even when you know that x is a multiple of 4, you can’t come up with a fixed value for x. Statement (2) by itself isn’t sufficient, so the answer can be only Choice (C) or (E). You still have one more evaluation: whether the two statements together provide sufficient information.

   Multiples of 4 are always even, so the two statements together don’t point you to the value of x. So the correct answer is Choice (E).

6. D. This question requires you to work with very large numbers, so you need to know what large numbers look like.

   One billion = 1,000,000,000, and 1 million = 1,000,000. In other words, 1 billion is 1,000 million.

   Now, look at the question at hand: 6.5 billion is written as 6,500,000,000. Writing out 6 billion is obvious, and 0.5 billion is one-half of 1,000 million, which is 500 million, or 500,000,000. You write 300 million like 300,000,000. To solve for the percentage, simply divide 300,000,000 by 6,500,000,000, using the fraction form:

   

   Simplify things by canceling out eight zeros on the top and bottom. (This step is legal because you’re just reducing your fraction.) Then divide 3 by 65.

   You don’t actually have to complete the mathematical calculation, because all the answer choices are derivatives of 46. You do need to know, though, that when you divide 3 by 65, your answer will have three places after the decimal. If you can’t figure this in your head, quickly set up the division problem on your noteboard and mark where the decimal will be in your answer.

   So , but the question asks for a percentage. To convert the decimal to a percentage, move the decimal point two places to the right and add a percentage sign. The answer is 4.6 percent.

   You can also use estimation to narrow down the answers. is about , which reduces to , or 0.05. The answer has to be slightly smaller than 0.05 because is less than , The answer that is slightly less than 0.05 is 0.046 or

7. B. To determine the value of , you have to figure out which of the four operations the symbol represents. The way to do so is to plug each of the operations into the equations offered by each of the two statements and see whether either of them allows you to narrow the symbol down to just one operation.

   Statement (1) gives you . Plug in each operation to see whether any make the equation true. You know addition and subtraction don’t work because you can’t add or subtract 1 to or from a number and end up with the same number. Both multiplication and division work: , and . So Statement (1) isn’t sufficient because it doesn’t allow you to narrow the symbol down to just one operation. The answer, then, can’t be Choice (A) or Choice (D).

   Statement (2) offers . The only difference between this equation and the one in Statement (1) is the answer. You know that multiplication and division don’t work, because they already produced an answer of 0. Subtraction results in –1, so the only operation that works is addition (0 + 1 = 1). This means that Statement (2) alone gives you enough information to determine which operation the symbol stands for, which allows you to figure out the value of .

   Data sufficiency questions don’t ask for the actual numeric answer, so don’t take the time to determine the actual value of the operation (not that it would take you long to do so for this question).

8. A. Evaluate each statement to determine whether it allows you to figure out the exact number of burrito sales for the day.

   You can construct a mathematical equation from the language in Statement (1). The unknown is the total number of today’s burrito sales. Let b = today’s burritos. Fewer means subtraction, so yesterday’s sales equal 2b – 100. The equation then looks like this:

   

   This equation has only one variable, so you know you can easily solve this equation to find out how many burritos left the shop today. (Don’t take the time to actually figure it out, though!) Statement (1) is sufficient, and the answer is either Choice (A) or Choice (D). To determine which it is, evaluate Statement (2).

   Statement (2) tells you that the number of burritos sold at Dave’s Wraps yesterday was 20 more than the number sold today, but this statement gives you two variables. You don’t know how many burritos were sold today, and you don’t know how many went out the door yesterday. If y stands for yesterday’s burrito sales, the equation would look something like this: y = 20 + b. You can’t definitively solve an equation with two variables without more information, so Statement (2) isn’t sufficient. The correct answer is Choice (A).

   (Oh, and if you won’t be able to sleep unless we confirm for you the number of burritos sold today, it’s 225: 450 = 2b, so 225 = b. Now be sure to get your sleep; you need it for the GMAT!)

9. B. The first thing that should jump out at you is that the first $1,000 of purchases is tax-free, so you don’t need to consider the first $1,000. Subtract $1,000 from $1,220 to get the value of purchases that will actually be taxed: $220.

   To find the amount of tax due, you multiply 220 by 7 percent (or 0.07), but you don’t have to take the time to fully work out the calculation. To make things simple, you can estimate: 200 is close to 220, and , so the amount has to be just a little more than $14.

   The only answer that’s just a little more than $14 is Choice (B). If you take the time to multiply 220 and 0.07, you’ll find that it’s exactly $15.40. But because this is a test where saving time is crucial, avoid making full calculations whenever possible.

10. D. The key to solving this problem is to recognize that a and b are supplementary angles, which means they add up to 180 degrees: . (Chapter 14 has more information on shapes and angles.)

    Now all you have to do is substitute 180 for a + b in the original equation and solve:

    

    So the correct answer is Choice (D).

11. A. To solve this problem, recognize that the halfway point between 100 and 75 is 87.5, so if x is greater than 87.5, it’s closer to 100. If it’s less than 87.5, it’s closer to 75. (If it equals 87.5, it’s the same distance from both.)

    If the difference between 100 and x (100 – x) is greater than the difference between x and 75 (x – 75), then x must be less than 87.5, because values greater than 87.5 would make 100 – x less than x – 75. Therefore, you absolutely know from Statement (1) that x is closer to 75. It’s sufficient to answer the question, and the answer is either Choice (A) or Choice (D).

    Now, look at Statement (2). Knowing that x > 85 doesn’t help, because values above 87.5 would make x closer to 100 and values between 85 and 87.5 would make it closer to 75. Statement (2) isn’t sufficient. For more about inequalities, consult Chapter 13.

12. B. This is a distance problem, so to determine the time of Ms. Nkalubo’s trip, you have to use the distance equation.

    The formula for distance is , which stands for (see Chapter 13 for details about this formula).

    Statement (1) is pretty easy to evaluate. Knowing that her average speed was 45 miles per hour gives you the rate value for the equation but nothing more, so you’re left with an unknown distance and an unknown amount of time. You can’t solve an equation with two variables without more information. Therefore, you can’t calculate her time. Statement (1) isn’t sufficient, so the answer can’t be Choice (A) or Choice (D).

    Statement (2) takes a little more thought. At first it may not appear to give you enough information to figure out time. But if you look further, you’ll see that it enables you to set up two simultaneous equations, and when you have two simultaneous equations with two variables, you can find the value of either variable. Here’s how: The first equation is for Ms. Nkalubo’s actual trip, which you can denote as Trip 1 (we’ve used a subscript 1 to show the values for Trip 1). Use the standard formula for distance:

    

    That’s as much as you know about Trip 1 for now.

    The second equation is for the theoretical trip proposed in the problem, which you can call Trip 2 (which we’ve denoted with a subscript 2). Start with the standard distance formula:

    

    Take the equation further with the information provided by Statement (2). Begin with the easy value. Trip 2 would take 3 hours, so . You also know that Ms. Nkalubo’s rate for Trip 2 was the rate of Trip 1. So . Substitute this value for rate into the equation for Trip 2:

    

    You should also recognize that d₁ and d₂ have the same value because the distances of the two trips are the same (it’s the same trip!). Therefore, you can set the left side of the first equation equal to the left side of the second and divide the rate variable from both sides.

    At this point, you have an equation with only one variable, so you know you can solve for the exact length of Ms. Nkalubo’s trip. Statement (2)’s information is sufficient to answer the question, so the correct answer is Choice (B).

    For those of you who hate to be left hanging and need to see how the equation turns out, we’ll finish the calculations. Just remember, you shouldn’t do this part for the test; it’s a waste of time. Here’s what the solution looks like:

    

    The time is hours, which is equal to 3 hours and 45 minutes. The family was probably ready for some action after almost four hours in the car!

13. E. Don’t let the language of this problem scare you. You’re really just applying basic operations.

    The arithmetic mean is 9.5 and the standard deviation is 1.5, so you’ll use a deviation of 1.5 to find values that stray from the mean. This means that the values that are 1 standard deviation from the mean are 11 and 8, which is the mean (9.5) plus or minus the standard deviation (1.5). The values that are 2 standard deviations from the mean are 12.5 and 6.5, which you get from adding and subtracting 3 () from the mean of 9.5. The values that are 3 standard deviations from the mean are 14 and 5, which you derive by adding and subtracting 4.5 () from the mean.

    So to solve this problem, you find that the values that are 2.5 standard deviations from the mean are 13.25 and 5.75, because . Look for an answer choice that’s more than 13.25 or less than 5.75. The answer is 13.5, Choice (E).

14. C. The four angles lie along a straight line, so they add up to 180 degrees. (If you need a refresher on the properties of angles, read Chapter 14.)

    Although it’s lovely to know that BX bisects (which means cuts exactly in half) the two angles on the left side and that BZ bisects the two angles on the right side, without the measure of at least one of the angles, you have no way of knowing the measurements of any of the angles. So Statement (1) isn’t sufficient, and the answer has to be Choice (B), Choice (C), or Choice (E).

    Statement (2) gives you only one of the angle measures, which by itself doesn’t clarify the measure of any better than Statement (1) does. Statement (2) isn’t sufficient.

    But remember that we said that for Statement (1) to work, you just need a value for at least one of the angles. Well, Statement (2) provides that value. Taken together, the two statements allow you to solve for the measure of . You can stop right there. The correct answer is Choice (C).

    You don’t have to actually figure out the measurement of the angle, but because we’re so thorough, we’re going to go through the calculations for you anyway. This step is unnecessary on test day. Knowing that BZ bisects and that measures 60 degrees allows you to deduce that is also 60 degrees. Additionally, you’ve now accounted for 120 of the total 180 degrees allotted for the four angles, which leaves you 60 degrees to play with. Finally, because BX bisects , two equal angles remain. Two equal angles that together equal 60 degrees must equal 30 degrees each, because .

15. D. To find the total distance of Traci’s trip, set up an equation that expresses the sum of the three separate trip portions. Let x equal the total distance in miles. Traci stopped to rest after she traveled of the total distance, so the first part of the trip is . She stopped again after she traveled of the distance remaining between her first stop and her destination, which is the total distance she traveled minus the first part of her trip. You can represent the second part of the trip mathematically, like this:
    

    The third part of the trip is the remaining 200 miles. Add up the three parts of the trip to set up the equation and solve for total distance:

    

    At this point, you can make it easier on yourself by multiplying each expression on both sides by 6 to get rid of the fractions:

    

    Traci traveled a total distance of 400 miles, so the correct answer is Choice (D).

16. E. This problem seems simple, but if you try to solve it too quickly, you may miss something. So consider all possibilities.

    Evaluating Statement (1) can be tricky. Don’t jump to the conclusion that if the lowest common denominator (LCD) of the two fractions is 10, then must have a denominator of 10 and, therefore, b = 10.

    The value of b could also equal 2, and the two fractions would still have an LCD of 10. Because b has two possible values, Statement (1) is insufficient. Therefore, the answer is Choice (B), Choice (C), or Choice (E).

    Statement (2) is easier to evaluate. The value of the numerator has no bearing on the value of the denominator, so the fact that a = 3 is irrelevant to the value of b. Statement (2) is also insufficient, which means the answer is either Choice (C) or Choice (E).

    Knowing that a = 3 tells you nothing about whether b is 10 or 2, which means that the two statements together are still insufficient to answer the question.

17. C. You could try to solve for n, but a faster and easier way to approach this problem is to plug each of the answer choices into the given equation and pick the one that doesn’t make the expression true:

    • Choice (A) gives you 1. Plug in 1 for x in the equation: . Doing so makes n = 1, which is a positive integer. Because 1 is a possible value for x, Choice (A) is wrong.
    • If you substitute 13 from Choice (B), you get . And 13 + 3 is 16 and . If n = 2, it’s a positive integer, so eliminate Choice (B).
    • For Choice (C), you substitute 45 into the equation: . The equation comes out to , and although it may seem like 4 could be a root of 48, it’s not. There’s no way n could be a positive integer when x = 45. Choice (C) is the correct answer. You can choose Choice (C) and go on, or you can check the last two answers just to be sure. Your decision depends on how much time you have remaining.
    • If you plug in 61 from Choice (D) into the equation, you get . And 61 + 3 = 64, which is . But 3 is a positive integer, so Choice (D) can’t be right.
    • Choice (E) is 253, and 253 + 3 = 256. And , which would make n = 4, a positive integer. Choice (E) makes the equation true, so it’s the wrong answer.

    Be careful when you answer questions that ask you to find the answer that can’t be true. In these cases, if an answer choice works, you have to eliminate it rather than choose it. Keep reminding yourself of your goal.

18. B. The first statement provides an equation that contains b, but notice that b is squared, so it’s likely the solution for b in this equation could be either positive or negative. You can perform a quick check to be sure. The left side of the equation equals 1, so the exponent must equal 0. Any value to the power of 0 is equal to 1. When you set the exponent equal to 0 and solve, you get two possible values for b:
    

    Statement (1) isn’t sufficient, so eliminate Choices (A) and (D).

    At first glance, Statement (2) appears insufficient as well because it contains more than one variable, but check it to be sure. First, make the bases equal:

    

    Then set the exponents equal to each other to discover whether you can solve for b:

    

    The minute you realize that the c values cancel, you know that you can solve for b. The answer must be Choice (B).

19. B. Apply the average formula to find the current month’s average number of automobiles sold. To find the sum, you need to add all the values represented on the plot. This stem-and-leaf plot presents a set of values in terms of their tens and ones digits. The left column is the tens digit, and the right column is the ones digit for each of the numbers of automobiles sold. Use data to find the sum:
    

    Enter this data into the average formula:

    

    At this point, you may notice that to obtain an average of 35 autos, each sales associate needs to sell 5 additional autos.

    If this isn’t obvious at first, you can apply the formula again to determine the total number of additional autos the sales associates need to sell next month to achieve an average of 35 autos sold per salesperson:

    

    This number is the total number of additional autos the sales associates need to sell next month to reach management’s goal, but the question asks for the average number each sales associate needs to sell to achieve an average of 35 autos per salesperson. So divide 110 by 22:

    

    On average, each sales associate needs to sell 5 additional automobiles to reach an average of 35 autos per salesperson.

20. A. This question requires basic simplification. Begin by canceling terms. Because the whole numerator is squared, you first need to factor every term in the parentheses. Take them one-by-one and apply process of elimination to the answer choices as you go.
    

    Divide the coefficients of 16 and 2 to get 8 and eliminate Choices (D) and (E) because they don’t have a coefficient of 8. Continue by squaring the variables in the numerator. When you take an exponent to another power, you multiply the exponents:

    
    

    The new expression is .

    Divide the variables by subtracting the exponents:

    
    

    Combine the components to get a final answer of .

    If you picked Choice (B), you subtracted the 2 from 6 when you worked with the b exponents. When you subtract a negative value, you actually add the value. If you picked Choice (C), you added the exponents when you squared them instead of multiplying them.

21. A. Notice the first statement gives you a ratio that contains both a and b. This looks promising. Set up the ratio by translating English to math: . It should be clear that you can solve this equation for , but if you want to sure, here’s this solution:
    

    Statement (1) is sufficient, so you eliminate Choices (B), (C), and (E) and move on to Statement (2). When you translate Statement (2) into a math equation, you get , which you can solve for a or b, but not for :

    

    Statement (2) isn’t sufficient, so Choice (D) is out, and Choice (A) is the answer.

22. C. Akhil begins with an initial CD investment of $1,200. Every six months he makes 1.05% on his existing money. So after the first six months, Akhil has 1.0105 times the $1,200 initial investment, or $1,212.60. But the balance doesn’t increase by $12.60 every six months because Akhil makes 1.05% on the new existing balance, So after one year, he has 1.0105 times $1,212.60 instead of $1,200, which is a balance of $1,225.33. At a year and a half, Akhil has 1.0105 times $1,225.33, or $1,238.20. Six months later the CD matures after another 1.05% is added to Akhil’s balance. .

    The amount Akhil invests into a second CD is $1,251.20 less the $400 he uses to purchase the tablet: . Pick Choice (C).

    You can save some time by making a comparison between what Ahkil earns from compounded interest to what he would earn from simple interest. The amount of compounded interest earned in a certain time period should be greater than that earned from simple interest. By multiplying $1,200 by 0.0105, you know he earns $12.60 each 6 months. Multiplying $12.60 times 4 give you $50.40 earned from simple interest in 2 years, which would make Ahkil’s balance $1,250.40. When you subtract $400 for the tablet, you learn that Ahkil would have $850.40 to invest in the second CD if he had earned simple interest. You can eliminate Choice (B) because he would have earned slightly more with compound interest but not so much more that you can justify Choices (D) or (E). Choice (C) is the only value that fits that description!

23. B. The equation to find slope is . Simply plug in values in this question. When you plug the values into the slope equation and solve, you get this:
    

    Pick Choice (B) and move on to the next question!

24. E. This is the last question, and it happens to be one of the most difficult ones of the bunch. At first, you may think that you can solve this question with two simultaneous equations. However, when you take a closer look, you see this isn’t the case. To get started, let f = the cost of a floor seat and b = the cost of a balcony seat. Then evaluate the statements.

    If you write out Matthew’s information in Statement (1) in mathematical terms, you get an equation with two variables:

    

    As we’ve said before, you can’t solve an equation with two variables without additional information. This statement alone isn’t sufficient, so the answer is either Choice (B), (C), or (E).

    Likewise, Statement (2)’s information leads to an equation with two variables:

    

    This equation alone isn’t enough to solve the problem, so the answer has to be Choice (C) or Choice (E).

    Here’s where you may have gotten prematurely excited. You may have thought that Statements (1) and (2) provided simultaneous equations that could be manipulated to give you the value of one of the variables. But if you look more closely, you’ll see that the equations are exactly the same. When you reduce the second equation or expand the first, you have identical equations. Look at the second equation:

    

    Divide both sides by 2:

    

    You don’t have simultaneous equations at all, and the two statements together won’t enable you to solve the problem. Mark Choice (E).

Be aware of data-sufficiency questions that ask you to find the sum (or difference, product, or quotient) of two variables rather than the individual value of one or both because the rule of thumb that two equations are needed to solve for two unknowns may not apply in such a case. If Matthew had purchased an equal number of floor seats and balcony seats, let’s say 4 of each, the equation for his information would have looked like this: . Since the questions asks for the value of , the equation for Matthew’s information would be sufficient by itself because you could solve for by factoring 4 from both sides of the equation.