CHAPTER 17

Math Formulas on the GMAT

If the average (arithmetic mean) of x, 25, y, and 30 is x + y, which of the following equals the value of x?

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Above is a typical Problem Solving question dealing with a commonly tested math formula. In this chapter, we’ll look at how to apply the Kaplan Method to this question, discuss the common math formulas tested on the GMAT, and go over the basic principles and strategies that you want to keep in mind on every Quantitative question involving math formulas. But before you move on, take a minute to think about what you see in this question and answer some questions about how you think it works:

• What math formula is being tested in this question?

   

• What does the format of the answer choices tell you about your task?

   

• What can you infer about how the GMAT constructs questions based on common math formulas?

   

• What GMAT Core Competencies are most essential to success on this question?

   

PREVIEWING MATH FORMULAS ON THE GMAT

What Math Formula Is Being Tested in This Question?

This question tests your understanding of the formula to solve for the average, or arithmetic mean, of a group of values. In this case, you are told the average (x + y) and are asked to solve for a missing term (x).

What Does the Format of the Answer Choices Tell You about Your Task?

The answer choices are algebraic expressions, not numerical values, so you can infer that the question must not supply enough information to solve for the actual value of x. Rather, what you will solve for is x “in terms of y.” All this means is that the actual value of x will vary depending on the value of y.

From the format of the answer choices, you can glean that your task on this question is to set up the average formula, plug in the information provided by the question stem into the appropriate positions in the formula, and use what you know about solving algebraic equations to isolate the variable x on one side and everything else on the other.

What Can You Infer About How the GMAT Constructs Questions Based on Common Math Formulas?

The GMAT assumes that you have a working knowledge of several classic formulas. You should be comfortable using these so that you don’t have to waste time trying to remember them on Test Day.

The GMAT tests these formulas by giving you some information that relates to a known formula and asking you for a different, missing piece of information that can be solved for using that formula.

Many of the formulas commonly tested on the GMAT have three parts. For example:

This chapter will cover the formulas listed above, plus others. The important thing to remember is that when you are given two parts of any three-part formula, you can always solve for the missing part. Keep this in mind when you encounter a question that seems to have you stumped. Always start by figuring out what formula is at issue; then figure out what parts of the given information can be substituted into that formula. At this point, the path to the solution will usually become clear.

What GMAT Core Competencies Are Most Essential to Success on This Question?

As with all questions that hinge on standard mathematical formulas, you must first use your Pattern Recognition skills to identify the relevant formula. Once you’ve done so, Critical Thinking becomes important as you determine how the given information applies to the formula at hand. The GMAT often presents information in a less-than-intuitive way, forcing you to decode the meaning and relevance of that information to the solution.

Here are the main topics we’ll cover in this chapter:

• Averages

• Rates and Speed—Converting Rates

• Rates and Speed—Multi-part Journeys

• Combined Rates and Combined Work

• Interest Rates

• Overlapping Sets

Handling GMAT Math Formulas

Now let’s apply the Kaplan Method to the math formulas question you saw earlier:

If the average (arithmetic mean) of x, 25, y, and 30 is x + y, which of the following equals the value of x?

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Step 1: Analyze the Question

This question gives you a lot of information. You are told that the average of the four terms, x, 25, y, and 30, is x + y. A quick glance at the answer choices reveals that you are being asked to solve for x in terms of y.

Step 2: State the Task

You’ll need to set up the average formula and then isolate x. But don’t rush to the conclusion that you have to do a lot of algebra here.

Step 3: Approach Strategically

Think critically about how you can approach this problem to save time. Picking Numbers is almost always a simple way to handle problems in which you need to solve for one variable in terms of another. Once you put the terms in the average formula, you can pick a number for x, solve the problem for y, and then plug that number into the answer choices to get the one that matches your choice for x. The average formula is the sum of terms divided by the number of terms. Here, that would look like this: . Multiplying both sides by 4 results in x + 25 + y + 30 = 4 (x + y). Distribute the 4 and you have x + 25 + y + 30 = 4x + 4y.

Now pick an easy number to substitute for x. Let’s have x = 1. That gives you 1 + 25 + y + 30 = 4 + 4y. Combine like terms, and you have 56 + y = 4 + 4y. Doing the subtraction necessary to get y and the numbers on different sides of the equation leaves you with 52 = 3y. Divide each side by 3, and you find that y = 17. Since you chose 1 to stand in for x, all you need to do is substitute 17 in for y and find the answer choice that equals 1. That’s choice (E).

You could also have solved algebraically. Taking the equation above, x + 25 + y + 30 = 4x + 4y, you can continue to simplify by combining like terms until you get 55 = 3x + 3y. Factoring out a 3 from the right side gives you 55 = 3(x + y), and dividing both sides by 3 results in = x + y. Isolating x and converting to a mixed number results in x = 18−y, which corresponds to choice (E).

Step 4: Confirm Your Answer

Using Picking Numbers can save you valuable time on this problem. You can also solve it with pure algebra, of course, and that approach can be used to confirm your answer if you initially solve by Picking Numbers.

Now let’s look at each of the common math formulas that show up on the GMAT Quantitative section, starting with averages.

AVERAGES

The average (arithmetic mean) of a group of numbers is defined as the sum of the values divided by the number of values.

Example: Henry buys three items costing $2.00, $0.75, and $0.25. What is the average price?

Balanced Average: You can save yourself a lot of laborious and error-prone calculation if you think of the average as a “balancing point” between the numbers in the series. That is, the difference between the average and every number below it will equal the difference between the average and every number above it. The “balanced average” approach dramatically reduces the difficulty of the arithmetic.

You’ve already seen that the average of $2.00, $0.75, and $0.25 is $1.00. Here’s how that balances:

Now let’s use this approach to solve a problem.

Example: The average of 43, 44, 45, and x is 45. What is the value of x?

Since the amount above the average must equal the amount below the average, the amount above must be 3. Therefore x is 3 above the average: 3 + 45 = 48; x = 48.

Weighted Average: A further way to calculate averages on the GMAT is to use the “weighted average” formula:

Weighted average of n terms = (Percent₁)(Average₁) + (Percent₂)(Average₂) + … + (Percent_n)(Average_n)

Weighted averages are useful when you know the average of different portions of the whole. For example, if two-fifths of the students in a class have a GPA of 79 and the remaining three-fifths have an average of 84, you could set up the weighted average formula as follows:

Notice the average of the whole class comes out to be closer to 84 than to 79. This will always be the case when the portions of the whole are of different sizes. In fact, this is where the term “weighted average” comes from; the larger the portion of the whole, the more heavily “weighted” that portion is when calculating the overall average. Note that you can use the weighted average formula as long as you have averages for all portions adding up to 100 percent of the whole.

In-Format Question: Averages on the GMAT

Now let’s use the Kaplan Method on a Data Sufficiency question dealing with averages:

If each of the bowlers in a tournament bowled an equal number of games, what is the average (arithmetic mean) score of all the games bowled in the tournament?

(1)   70 percent of the bowlers had an average (arithmetic mean) score of 120, and the other 30 percent had an average score of 140.

(2)   Each of the 350 bowlers in the tournament bowled 3 games.

Step 1: Analyze the Question Stem

First, determine the type of Data Sufficiency question you’re dealing with. You need one exact value for the average score of the games, so this a Value question.

There’s no direct simplification to be done, but the fact that you’re asked for an average should alert you to the possibility that you can get the answer in a variety of ways—either through direct calculation of the scores or through a weighted average approach.

To answer the question, you’ll need either the number of games and the total of the scores or some way to calculate a weighted average.

Step 2: Evaluate the Statements Using 12TEN

Statement (2) is very straightforward, so starting there makes sense. This tells you the number of games but nothing about the scores. Insufficient. Eliminate (B) and (D).

Statement (1) doesn’t allow you to figure out exactly the number of games or the exact sum of the scores. But since the proportions add up to 100 percent of the total, and since all the bowlers bowled the same number of games, you can calculate the overall average using the weighted average approach. Sufficient. The answer is (A).

(Remember, you don’t actually want to calculate the average, just know that you could. That calculation would be Overall average = 0.7(120) + 0.3(140).)

The GMAT rewards those who think critically and find the most efficient approach to a problem. By using the weighted average approach, you avoided messy calculations and saved valuable time.

TAKEAWAYS: AVERAGES

 

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• Balanced average approach: the sum of the differences between each term and the average must be zero. You can solve some average questions using this fact alone.

• Weighted average of n terms = (Percent₁)(Average₁) + (Percent₂)(Average₂) + … + (Percent_n)(Average_n)

Practice Set: Averages on the GMAT

Answers and explanations at end of chapter

1. Jerry’s average (arithmetic mean) score on the first 3 of 4 tests is 85. If Jerry wants to raise his average by 2 points, what score must he earn on the fourth test?

   • 87

   • 89

   • 90

   • 93

   • 95

2. The average (arithmetic mean) of all scores on a certain algebra test was 90. If the average of the 8 male students’ grades was 87, and the average of the female students’ grades was 92, how many female students took the test?

   • 8

   • 9

   • 10

   • 11

   • 12

3. An exam is given in a certain class. The average (arithmetic mean) of the highest score and the lowest score on the exam is equal to x. If the average score for the entire class is equal to y and there are z students in the class, where z > 5, then in terms of x, y, and z, what is the average score for the class, excluding the highest and lowest scores?

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4. If the average (arithmetic mean) of four numbers is 10, how many of the numbers are greater than 10?

(1)   Precisely 2 of the numbers are equal to 10.

(2)   The largest of the 4 numbers is 10 greater than the smallest of the 4 numbers.

RATES AND SPEED—CONVERTING RATES

You’ve seen this formula before:

The most common rate is a speed—miles per hour—but anything with the word per is a rate: kilometers per second, miles per gallon, ounces of cheese per party guest, and so forth. A rate is any quantity of A per quantity of B.

Most rate problems involve conversions from one rate to another. These are best handled by multiplying the various rates so that measurements cancel. You may have to invert some of the rates to make the measurements cancel.

Example: If a car averages 25 miles per gallon and each gallon of gas costs $2.40, what’s the value of the gas consumed in a trip of 175 miles?

Here’s what a GMAT question might look like that gives you a rate that you must then convert. Follow along with this example:

If José reads at a constant rate of 2 pages every 5 minutes, how many seconds will it take him to read N pages?

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• 2N

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• 24N

• 150N

There are two common ways of solving rate problems: algebraically and by using the Picking Numbers strategy. Let’s try both for the problem above.

Algebraic Solution: You can call the number of seconds that you’ve been asked to find T. The first hurdle that you need to clear in this problem is that you’re given a rate in minutes but asked to calculate the number of seconds. Change the rate you’re given to seconds so you can get that aspect of the problem out of the way. There are 60 seconds to a minute. So you need to change the rate as follows:

Since the rates are the same, N and T will be in the same proportion as the rate you are given. So you can set up this equation:

Now cross multiply to solve for T: T = 150N. The answer is (E).

Picking Numbers Solution: Since this question asks you to solve for the amount of time it takes José to read an unspecified number of pages, you can make the situation more concrete by picking a number for N. Keep the number simple, so that it’s easy to work with: try N = 2, since José reads exactly 2 pages every 5 minutes, or 300 seconds (5 × 60 seconds = 300 seconds). Now look for the answer choices that yield 300 when N = 2. The only one that does so is (E), so that must be the answer.

In-Format Question: Rates and Speed—Converting Rates on the GMAT

Now let’s use the Kaplan Method on a Problem Solving question dealing with converting rates:

If the city centers of New York and London are 3,471 miles apart, which of the following is closest to the distance between the city centers in inches? (There are 5,280 feet in a mile.)

• 1.75 × 10⁷

• 1.83 × 10⁷

• 2.10 × 10⁸

• 2.10 × 10⁹

• 2.20 × 10¹⁰

Step 1: Analyze the Question

There are some scary numbers in this question. Attention to the Right Detail can help you zero in on the most crucial piece of information: the words closest to let you know that you can afford to estimate, as does the fact that most answers are a whole power of 10 different from each other.

Step 2: State the Task

Your task is to convert 3,471 miles into inches.

Step 3: Approach Strategically

The problem gives you 5,280 feet per mile, allowing you to convert miles to feet. (The GMAT will expect you to know more common conversions, such as 12 inches per foot.)

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That’s 3,471 × 5,280 × 12. Happily, you can estimate, so save time by changing that to 3,500 × 5,000 × 10 = 175,000,000, or 1.75 × 10⁸. The only answer that’s remotely close (within a power of 10) is (C), so (C) is the answer.

Step 4: Confirm Your Answer

Reread the question stem, making sure that you didn’t miss anything about the problem. For example, if you accidentally solved for feet instead of inches, you’d have chosen (A) or (B). This step would save you from the error.

TAKEAWAYS: RATES AND SPEED—CONVERTING RATES

 

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• When solving a conversion algebraically, set up the equation so that units cancel.

• Keep track of the units of measurement. Rate questions often involve converting minutes to hours or days to weeks.

Practice Set: Rates and Speed—Converting Rates on the GMAT

Answers and explanations at end of chapter

5. A truck owner will refuel his vehicle at a rate of 2 gallons of diesel fuel every 5 seconds. If diesel fuel costs $1.25 per gallon, how long will it take for the truck owner to refuel his vehicle with $40 worth of diesel fuel?

   • 20 seconds

   • 1 minute

   • 1 minute 20 seconds

   • 20 minutes

   • 1 hour 20 minutes

6. The moon revolves around the earth at a speed of approximately 1.02 kilometers per second. This approximate speed is how many kilometers per hour?

   • 60

   • 61.2

   • 62.5

   • 3,600

   • 3,672

7. Magnabulk Corp sells boxes holding d magnets each. The boxes are shipped in crates, each holding b boxes. What is the price charged per magnet, in cents, if Magnabulk charges m dollars for each crate?

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RATES AND SPEED—MULTI-PART JOURNEYS

The GMAT often uses scenarios involving multi-part journeys in order to ask questions about “average rates.” In such cases, the testmakers will give you information that includes two or more different rates or speeds. So far in this chapter, you’ve learned about both averages and rates, so it’s understandably tempting in this situation simply to average the different individual rates you’re given.

Don’t fall for this trap. Just as you cannot use the averages of two different portions of a group to determine the group’s overall average (unless you know the weights of the portions), you also cannot find the average speed of a journey from the speeds of two parts unless you know the proportion of time spent or distance traveled in those parts. The GMAT’s biggest trap with multi-part journeys or tasks is to offer an answer choice that fails to account for the weighting of the averages and simply splits the difference between the two portions. The GMAT writes its problems so that this is never the correct solution. Instead, you’ll need to use the average rate formula.

Average Rate (Average A per B):

Example: John travels 30 miles in 2 hours and then 60 miles in 3 hours. What is his average speed in miles per hour?

Notice how you can check your work on this question by applying some Critical Thinking: John’s first rate was 15 miles per hour, and his second rate was 20 miles per hour, so his average rate will fall somewhere between 15 and 20 miles per hour. You can estimate effectively on this question, as on most average rate questions, by identifying the rate at which John spent longer traveling. In this case, John spent more time traveling at 20 miles per hour, so his average rate should be closer to 20 than to 15. Indeed, 18 miles per hour is consistent with the direction in which the average rate should be weighted.

Average rate problems on the GMAT can become quite complex. If your paraphrase of the question stem is anything like “First this person moves at one speed, then at another,” then you’re dealing with a multi-part journey problem. There is frequently a lot of information to deal with.

To organize the data on complicated multi-part journey problems, try jotting down this chart on your noteboard:

	Rate	Time	Distance
part 1 of trip
part 2 of trip
entire trip

This chart folds many equations into one:

• Rate × Time = Distance; each row of boxes multiplies across.

• Time (part 1) + Time (part 2) = Time (entire trip); time boxes add down.

• Distance (part 1) + Distance (part 2) = Distance (entire trip); distance boxes add down.

• Rates do not add down; they are calculated by dividing distance by time. If you need to calculate the speed for the entire trip, use the average speed formula:

Let’s go through a sample question, and you’ll see how the chart helps you.

A powerboat crosses a lake at 18 miles per hour and returns at 12 miles per hour. If the time taken turning the boat around was negligible and it returns by the same route, then what was the boat’s average speed for the round trip, in miles per hour?

When you notice that you have a multi-part journey problem, jot down the chart, using a question mark to indicate the value that the question asks you for:

	Rate	Time	Distance
part 1 of trip
part 2 of trip
entire trip	?

Now fill in the data given by the problem:

	Rate	Time	Distance
part 1 of trip	18
part 2 of trip	12
entire trip	?

Now you clearly have to get some information about either the time or the distance to be able to solve for much of anything. The only other piece of information you have is that the boat traveled the same route both times. That means the distance must have been the same. You could use the variable D, but the math will get fairly complicated if you do. Instead, keep it simple and Pick Numbers. A number that’s a multiple both of 18 and of 12 will make the math work out nicely, so let’s say the lake was 36 miles across:

	Rate	Time	Distance
part 1 of trip	18		36
part 2 of trip	12		36
entire trip	?

Now you can solve for the times and the total distance:

	Rate	Time	Distance
part 1 of trip	18	2	36
part 2 of trip	12	3	36
entire trip	?		72

And finally for total time:

	Rate	Time	Distance
part 1 of trip	18	2	36
part 2 of trip	12	3	36
entire trip	?	5	72

Now you have what you need to solve for average rate. It’s mph. This answer makes logical sense, since the two rates are 18 and 12 miles per hour and the boat spent more time traveling at the lower speed, so the average is closer to 12 than to 18.

In-Format Question: Rates and Speed—Multi-part Journeys on the GMAT

Now let’s use the Kaplan Method on a Data Sufficiency question dealing with a multi-part journey:

What was the average speed of a runner in a race from Point X to Point Z?

(1)   The runner’s average speed from Point X to Point Y was 10 miles per hour.

(2)   The runner’s average speed from Point Y to Point z was 8 miles per hour.

Step 1: Analyze the Question Stem

This is a Value question, so you need one exact speed for the trip from Point X to Point Z.

Use Critical Thinking: there’s no direct simplification to be done, but since you can sometimes calculate exact speeds with unknown distance or time, there might be a way to solve for speed without knowing the exact distance and time.

To answer the question, you’ll need either the exact distance and time or some information about distance or time that allows you to calculate speed.

Step 2: Evaluate the Statements Using 12TEN

Statement (1) gives you some data about the trip from X to Y but nothing about X to Z. Insufficient. Eliminate (A) and (D). Likewise, Statement (2) gives you data about the trip from Y to z but nothing about X to Z. Insufficient. Eliminate (B).

Now you must combine. Notice that something crucial is missing—information relating either the times or the distances of the X-to-Y and the Y-to-Z leg. If the X-to-Y distance were 10 miles and the Y-to-Z distance were 10 inches, the average speed would be essentially 10 mph. And if it’s the other way around, with the X-to-Y distance 10 inches and the Y-to-Z distance 10 miles, the average speed would be essentially 8 mph. The combined statements are insufficient, so the answer is (E).

TAKEAWAYS: RATES AND SPEED—MULTI-PART JOURNEYS

 

• In average rate and average speed questions, one wrong answer choice involves not accounting for the weighting of the averages. To avoid this wrong answer, use the average rate and average speed formulas.

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• For multi-part journeys, use a chart to organize data.

• In an average speed question, if you recognize the direction in which the speed should be weighted, you can rapidly eliminate incorrect answers.

Practice Set: Rates and Speed—Multi-part Journeys on the GMAT

Answers and explanations at end of chapter

8. A motorcyclist started riding at highway marker A, drove 120 miles to highway marker B, and then, without pausing, continued to highway marker C, where she stopped. The average speed of the motorcyclist, over the course of the entire trip, was 45 miles per hour. If the ride from marker A to marker B lasted 3 times as many hours as the rest of the ride, and the distance from marker B to marker C was half of the distance from marker A to marker B, what was the average speed, in miles per hour, of the motorcyclist while driving from marker B to marker C?

   • 40

   • 45

   • 50

   • 55

   • 60

9. A canoeist paddled upstream at 10 meters per minute, turned around, and drifted downstream at 15 meters per minute. If the distance traveled in each direction was the same, and the time spent turning the canoe around was negligible, what was the canoeist’s average speed over the course of the journey, in meters per minute?

   • 11.5

   • 12

   • 12.5

   • 13

   • 13.5

10. A truck driver drove for 2 days. On the second day, he drove 3 hours longer and at an average speed of 15 miles per hour faster than he drove on the first day. If he drove a total of 1,020 miles and spent 21 hours driving during the 2 days, what was his average speed on the first day, in miles per hour?

    • 25

    • 30

    • 35

    • 40

    • 45

11. Did Jon complete a journey of 40 kilometers in less time than it took Ann to complete the same journey?

(1)   Jon traveled at an average speed of 30 kilometers per hour for the first 10 kilometers and then at an average speed of 15 kilometers per hour for the rest of the journey.

(2)   Ann traveled at an average speed of 20 kilometers per hour for the entire journey.

COMBINED RATES AND COMBINED WORK

Combined work questions present you with information about different people or machines that can perform the same job in different amounts of time. Rates in combined work questions are expressed in a similar way to how you’ll see them expressed in other rate questions:

, or number of tasks per amount of time

Combined work questions would be much more straightforward if they provided you with the rates at which people or machines work; instead, the testmakers make things more complicated by giving you the information in terms of how much time it takes for each individual to complete a given task. To calculate the total time needed for everyone working together to finish the job, use this formula:

The reciprocal of the time it takes everyone working together = the sum of the reciprocals of the times it would take each working individually. (The reciprocal of A is .)

Written algebraically, this formula comes out as follows:

, where A, B, C, etc. represent the time it takes the individual people or machines to do the job by themselves.

You can use this general version of the formula with any number of people or machines working together.

If, however, the question presents you with only two people or machines working together, you will save time by using the following simplified version of the combined work formula:

Again, A and B are the times it takes the two people working alone to finish the job. By solving for the total time directly instead of for its reciprocal, you’ll avoid a common trap.

Example: Bob can clean a room in 3 hours, and George can clean the same room in 2 hours. How many hours does it take Bob and George to clean the room if they work together but independently?

Total time = hours, or 1 hour 12 minutes.

Attention to the Right Detail is crucial on combined work problems in which the GMAT can give you the needed information in various forms. The Critical Thinker will be rewarded by seeing what the question requires and choosing the most strategic approach.

Let’s now look at a seemingly complicated GMAT work problem and see how the formula makes things much easier.

Working together, John, David, and Roger require 2 hours to complete a certain task, if each of them works at his respective constant rate. If John alone can complete the task in 4 hours and David alone can complete the task in 9 hours, how many hours would it take Roger to complete the task, working alone?

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• 9

• 12

Combined Work Formula: Here’s the solution, employing the combined work formula.

Hour-by-Hour Approach: An alternative approach to this kind of problem that some find intuitive and quick is to break down the work on an hour-by-hour basis. Take David first. By himself, he could do the entire task in 9 hours. Therefore, during every single hour that these three people work together, David will complete of the task ( is just the reciprocal of 9). In the second hour, he’ll do another . And in that extra hour, he’ll do of the task. Add them up: David will be doing of the entire task during the period in question.

How about John? He’s much faster than David. Working alone, he could complete the entire task in 4 hours (or hours). So during each hour he works, he’ll do the reciprocal of (or ) of the task. Multiply that of a task per hour by the 2 hours the three men work, and you see that John himself will account for or of the task.

With David and John accounting for and of the task respectively, that leaves exactly of the task to be performed by Roger. You can either see that this means that Roger and David work at the same rate—so it’ll also take Roger 9 hours—or just divide 2 hours by task to get 9 hours per task, choice (D).

Some harder work problems involve tasks left undone. In those cases, you’d need to use this hour-by-hour approach, so it’s worthwhile to be familiar with it.

In-Format Question: Combined Rates and Combined Work on the GMAT

Now let’s use the Kaplan Method on a Problem Solving question dealing with combined rates and combined work:

Working alone, machine X can manufacture 1,000 nails in 12 hours. Working together, machines X and Y can manufacture 1,000 nails in 5 hours. How many hours does it take machine Y to manufacture 1,000 nails working alone?

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• 7

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Step 1: Analyze the Question

You have a choice of approaches for this combined work question. You can add the rates or solve with the times directly, whichever you find easier.

Step 2: State the Task

You’re asked for the time it takes Machine Y to make 1,000 nails.

Step 3: Approach Strategically

Many combined work questions have answer choices that can be logically eliminated, and this is no different. Since the machines take five hours together, it’s hardly reasonable to think that one machine would take less than five hours on its own. (A) can’t be correct. Some estimation allows you to eliminate one more—choice (B) is very, very close to the combined time. Even though machine X isn’t very fast, it’s not so slow that it would save only a third of an hour. You’ve just eliminated two answer choices through Critical Thinking.

Since you’re given times and they are all for the same-sized task (manufacturing 1,000 nails), and the task is completed, the formula will apply nicely.

The answer is (E).

You also could have solved this problem by adding the rates.

If the tasks are all the same, you can save yourself a lot of work by just giving one task the value of 1—in this problem, meaning “1 batch of 1,000 nails.”

Step 4: Confirm Your Answer

Reread the question stem, making sure that you didn’t miss anything about the problem. More complicated combined work problems will shift the size of the task halfway through the question stem.

TAKEAWAYS: COMBINED RATES AND COMBINED WORK

 

• When using rates in a combined work question, .

• The combined work formula for two workers is , where T equals the time both workers take to do the task when working together, and A and B equal the time needed by each worker to do the job on his or her own, respectively.

• If more than two workers work together, use the generic formula for combined work:

Practice Set: Combined Rates and Combined Work on the GMAT

Answers and explanations at end of chapter

12. Kathleen can paint a room in 3 hours, and Anthony can paint an identical room in 5 hours. How many hours would it take Kathleen and Anthony to paint both rooms if they work together at their respective rates?

    • 

    • 

    • 

    • 

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13. Train A left Centerville Station, heading toward Dale City Station, at 3:00 p.m. Train B left Dale City Station, heading toward Centerville Station, at 3:20 p.m. on the same day. The trains rode on straight tracks that were parallel to each other. If Train A traveled at a constant speed of 30 miles per hour and Train B traveled at a constant speed of 10 miles per hour, and the distance between the Centerville Station and Dale City Station is 90 miles, when did the trains pass each other?

    • 4:45 p.m.

    • 5:00 p.m.

    • 5:20 p.m.

    • 5:35 p.m.

    • 6:00 p.m.

14. Truck X is 13 miles ahead of Truck Y, which is traveling the same direction along the same route as Truck X. If Truck X is traveling at an average speed of 47 miles per hour and Truck Y is traveling at an average speed of 53 miles per hour, how long will it take Truck Y to overtake and drive 5 miles ahead of Truck X?

    • 2 hours

    • 2 hours 20 minutes

    • 2 hours 30 minutes

    • 2 hours 45 minutes

    • 3 hours

INTEREST RATES

Interest rates are something that you might be acutely aware of in real life, especially as they apply to mortgages and credit cards, but unless you work directly in a field that touches on loans or mortgages, you probably don’t think about the formulas used to calculate them.

Fortunately, you don’t have to puzzle over calculating out the interest year-by-year or step-by-step. You can memorize and apply formulas to help you manage the information from the question stem much more efficiently and accurately than by simply applying brute force to the arithmetic.

Interest rate questions rely on three important pieces of information: (1) the principal, or the amount initially invested; (2) the interest rate, or the rate at which the investment grows; and (3) the time period during which the investment accrues interest.

Attention to the Right Detail is necessary even when applying the formulas—and no detail is more important to interest rate questions than whether the interest is simple or compound. Simple and compound interest use different formulas:

Simple interest is interest applied only to the principal, not to the interest that has already accrued. Use this formula:

(Total of principal and interest) = Principal × (1 + rt), where r equals the interest rate per time period expressed as a decimal and t equals the number of time periods

Example: If $100 were invested at 12 percent simple annual interest, what would be the total value of the investment after three years?

Total = $100 × (1 + 0.12 × 3) = $100(1.36) = $136

Compound interest is interest applied to the principal and any previously accrued interest. Use this formula:

(Total of principal and interest) = Principal × (1 + r)^t, where r equals the interest rate per time period expressed as a decimal and t equals the number of time periods

Example: If $100 were invested at 12 percent interest compounded annually, what would be the total value of the investment after three years?

Total = $100 × (1 + 0.12)³ = $100(1.12)³

Despite the GMAT’s general preference for simplified values in the answer choices, you will often see answer choices for a compound interest rate problem written as expressions similar to the result above. You can be thankful that you won’t have to calculate values such as 1.12³, since you don’t have the use of a calculator on the Quant section.

On Test Day you might also encounter a more difficult question that requires dealing with annual interest but payments not on an annual basis. In this case, r equals the annual rate divided by the number of times per year it is applied, and t equals the number of years multiplied by the number of times per year interest is applied.

Example: If $100 were invested at 12 percent annual interest, compounded quarterly, what would be the total value of the investment after three years?

In-Format Question: Interest Rates on the GMAT

Now let’s use the Kaplan Method on a Problem Solving question dealing with interest rates:

The number of bacteria in a petri dish increased by 50 percent every 2 hours. If there were 108 million bacteria in the dish at 2:00 p.m., at what time were there 32 million bacteria in the dish?

• 6:00 p.m.

• 8:00 p.m.

• 6:00 a.m.

• 8:00 a.m.

• 10:00 a.m.

Step 1: Analyze the Question

This may not look like an interest rate question at first glance, but interest rates are just percent increases applied multiple times. The number of bacteria increases 50% every 2 hours. You could think of this as 50% compounded interest applied once every 2 hours.

Step 2: State the Task

Instead of calculating forward, you have to calculate back—if there are 108 million in the dish at 2 p.m., when were there 32 million?

Step 3: Approach Strategically

This problem could be solved as a straightforward percent change, but let’s look at it through the lens of interest rates:

Clearly, then t = 3. That’s three 2-hour increases for a total of 6 hours. Since there were 108 million at 2 p.m., there were 32 million 6 hours earlier at 8 a.m.

The answer is (D).

This is also an excellent Backsolving question, as you could easily just try out a time for 32 million and see whether that’s consistent with 108 million at 2 p.m.

Let’s say you try (B) first:

Time	Millions of bacteria
8 p.m.	32
10 p.m.	48
midnight	72
2 a.m.	108

That’s either far too soon (12 hours too soon) or far too late (12 hours too late), depending on how you envision the day. Either way, (B) is wrong by a lot, and just shifting 2 hours to 6 p.m. won’t help matters. Eliminate (A) as well.

Now let’s try (D):

Time	Millions of bacteria
8 a.m.	32
10 a.m.	48
noon	72
2 p.m.	108

Exactly what it should be (you might also have seen that (D) would be correct when you noticed that (B), 8 p.m., was off by exactly 12 hours). The answer is (D).

Step 4: Confirm Your Answer

Reread the question stem, making sure that you didn’t accidentally skip over any important aspects of the situation.

TAKEAWAYS: INTEREST RATES

 

• Simple interest:

  (Total of principal and interest) = Principal × (1 + rt), where r equals the interest rate per time period expressed as a decimal and t equals the number of time periods.

• Compound interest:

  (Total of principal and interest) = Principal × (1 + r)^t, where r equals the interest rate per time period expressed as a decimal and t equals the number of time periods.

Practice Set: Interest Rates on the GMAT

Answers and explanations at end of chapter

15. If John invested $1 at 5 percent interest compounded annually, the total value of the investment, in dollars, at the end of 4 years would be

    • (1.5)⁴

    • 4(1.5)

    • (1.05)⁴

    • 1 + (0.05)⁴

    • 1 + 4(0.05)

16. The amount of an investment will double in approximately 70/p years, where p is the percent interest, compounded annually. If Thelma invests $40,000 in a long-term CD that pays 5 percent interest, compounded annually, what will be the approximate total value of the investment when Thelma is ready to retire 42 years later?

    • $280,000

    • $320,000

    • $360,000

    • $450,000

    • $540,000

17. A certain account pays 1.5 percent compound interest every 3 months. A person invested an initial amount and did not invest any more money in the account after that. If after exactly 5 years, the amount of money in the account was T dollars, which of the following is an expression for the original number of dollars invested in the account?

    • (1.015)⁴T

    • (1.015)¹⁵T

    • (1.015)²⁰T

    • 

    • 

OVERLAPPING SETS

Another classic GMAT setup involves a large group that is subdivided into two potentially overlapping subgroups. For example, let’s say that in a room of 20 people, there are 12 dog owners and 14 cat owners. Since 12 plus 14 is more than 20, the only way this situation makes any sense is if some people own both a dog and a cat. And it’s possible that some own neither. Essentially there are four different subgroups to consider: (1) those who own a dog but not a cat, (2) those who own a cat but not a dog, (3) those who own both a cat and a dog, and (4) those who own neither a cat nor a dog. You could also combine some of these groups to consider both the total number of dog owners and the total number of cat owners.

There are three ways to work these problems. Let’s look at each.

Approach 1: Overlapping Sets Formula

Use the overlapping sets formula:

Group 1 + Group 2 − Both + Neither = Total

Example: An office manager orders 27 pizzas for a party. Of these, 15 have pepperoni, and 10 have mushrooms. If 4 pizzas have no toppings at all, and no other toppings are ordered, then how many pizzas were ordered with both pepperoni and mushrooms?

Approach 2: Venn Diagram

Organize the given information using a Venn diagram. This approach uses partially overlapping circles to represent the data visually:

Example: All the students in a class study either Forensics or Statistics. There are 16 Forensics students, 3 of whom also study Statistics. How many Statistics students are in the class if the class has 25 students altogether?

Because 3 of the Forensics students also study Statistics, there must be 13 who study only Forensics. If the “Forensics only” and “Both” groups total 16 and there are 25 students in the class, then 25 − 16, or 9, students study only Statistics. Thus, 9 + 3 = 12, the total number of Statistics students. Putting all these numbers into the Venn diagram as you go helps you to see these relationships clearly:

Approach 3: Chart

Organize the given information using a chart. This way works best with complicated overlapping sets problems because it has a separate place for each of the nine data points you might be given. Take your time organizing the chart, and the problem will almost solve itself:

	in Group 1	not in Group 1	Total
in Group 2
not in Group 2
Total

Example: A company has 200 employees, 90 of whom belong to a union. If there are 95 part-time nonunion employees and 80 full-time union employees, then how many full-time employees are in the company?

Start by putting the data into the chart, using a question mark to indicate the value that the question asks for:

	in union	not in union	Total
full-time	80		?
part-time		95
Total	90		200

Now you can calculate the total number of nonunion employees (200 − 90 = 110) and the number of part-time union employees (90 − 80 = 10):

	in union	not in union	Total
full-time	80		?
part-time	10	95
Total	90	110	200

And now either calculate full-time nonunion employees (110 − 95 = 15) or the total number of part-time employees (10 + 95 = 105):

	in union	not in union	Total
full-time	80	15	?
part-time	10	95	105
Total	90	110	200

Either way, you can then calculate the total number of full-time employees (80 + 15 = 95 or 200 − 105 = 95):

	in union	not in union	Total
full-time	80	15	95
part-time	10	95	105
Total	90	110	200

There are 95 full-time employees.

Each of the three approaches has its plusses and minuses. Your own personal thinking style will respond best to one of these over the others. Practice all of them so you get a sense of which approach you like with different problems.

Now try the following example on your own, using Critical Thinking to assess which of the three approaches will be most effective:

A group of 25 children went to the circus, 60% of whom liked the clowns. If the number of boys who liked the clowns was three more than the number who didn’t, and the number of boys was one larger than the number of girls, then how many girls did not like the clowns?

In your overview of the problem, you can see that there are some complex relationships going on; you shouldn’t try to understand them all at once. But notice that there are two ways in which children can be classified: boy or girl, liked or didn’t like the clowns. That makes this an overlapping sets problem.

The approach that offers the most flexibility, shows the most detail, and is therefore frequently the safest to use with this question type is this chart:

	in Group 1	not in Group 1	Total
in Group 2
not in Group 2
Total

In this problem, the two groupings are boy or girl and likes clowns or doesn’t like clowns. So here’s the chart you’d use for this problem:

	likes clowns	doesn’t like	Total
boys
girls
Total

Now enter the information from the question stem:

	likes clowns	doesn’t like	Total
boys	x + 3	x	g + 1
girls		?	g
Total	60% of 25		25

Two calculations can be done right away: 60% of 25 = (0.6)(25) = 15. Also, (g + 1) + g = 25. That means 2g + 1 = 25, so 2g = 24 and g = 12.

	likes clowns	doesn’t like	Total
boys	x + 3	x	13
girls		?	12
Total	15		25

The next two calculations suggested by the chart are the total number of children who don’t like clowns, 25 − 15 = 10, and the value of x, (x + 3) + x = 13 or 2x + 3 = 13. That means 2x = 10, or x = 5.

	likes clowns	doesn’t like	Total
boys	8	5	13
girls		?	12
Total	15	10	25

Now you know the answer to the question … the number of girls who didn’t like the clowns is 10 − 5, or 5. You can fill in the number of girls who like clowns, too, just for fun.

	likes clowns	doesn’t like	Total
boys	8	5	13
girls	7	5	12
Total	15	10	25

Because this chart shows all nine possible data points in these problems, it allows you to answer any question that might be asked. Fraction of girls who liked the clowns? It’s . Percentage of children who didn’t like the clowns that were boys? It’s 50%. Ratio of girls who liked the clowns to boys who didn’t? It’s 7:5.

When overlapping sets questions involve proportions, be very clear about what the basis of the proportion is. For example, the number of boys who didn’t like the clowns could be described as 20% (of the children), 50% (of the children who didn’t like clowns), or 100% (as large as the number of girls who didn’t like the clowns).

In-Format Question: Overlapping Sets on the GMAT

Now let’s use the Kaplan Method on a Problem Solving question dealing with overlapping sets:

A polling company surveyed a certain country, and it found that 35% of that country’s registered voters had an unfavorable impression of both of that state’s major political parties and that 20% had a favorable impression only of Party A. If one registered voter has a favorable impression of both parties for every two registered voters who have a favorable impression only of Party B, then what percentage of the country’s registered voters have a favorable impression of both parties (assuming that respondents to the poll were given a choice between favorable and unfavorable impressions only)?

• 15

• 20

• 30

• 35

• 45

Step 1: Analyze the Question

You’re presented with a lot of information in this overlapping sets question. Don’t try to digest it all at once. Notice that the total group (registered voters) can be separated into two major categories (those who like Party A and those who like Party B) and that those categories are not mutually exclusive (which means that someone could be in both categories—in this case, that someone could like both parties). That’s the general setup for overlapping sets questions.

Use Critical Thinking to choose the best approach for this problem. This is a complicated question with many data points. The most powerful tool for understanding confusing or complicated overlapping sets is a chart:

	favorable B	not favorable B	Total
favorable A
not favorable A
Total

Step 2: State the Task

You need to calculate the percentage of registered voters who like both parties; in other words, the value in the upper left-hand box.

Step 3: Approach Strategically

Start by putting the information into the chart. For simplicity’s sake, pick the total number of voters to be 100.

	favorable B	not favorable B	Total
favorable A		20
not favorable A		35
Total			100

The other piece of data is “one registered voter has a favorable impression of both parties for every two registered voters who have a favorable impression only of Party B.” In other words, the ratio of “favorable A and favorable B” to “favorable B and not favorable A” is 1:2. So you have something else to put in the chart—let’s call that x and 2x.

	favorable B	not favorable B	Total
favorable A	x	20
not favorable A	2x	35
Total			100

Since this chart adds down (and across), you can fill in the rest easily enough:

	favorable B	not favorable B	Total
favorable A	x	20	20 + x
not favorable A	2x	35	35 + 2x
Total	3x	55	100

Whether you use the Total column or the Total row, you have the same equation to solve for x:

The answer is (A).

Step 4: Confirm Your Answer

Reread the question stem, making sure that you didn’t accidentally misread anything.

TAKEAWAYS: OVERLAPPING SETS

 

• Overlapping sets questions on the GMAT require you to determine the elements in a set that do, or do not, overlap with elements of other sets.

• There are three approaches for overlapping sets questions:

  • Use the overlapping sets formula: Total = Group 1 + Group 2 − Both + Neither.

  • Draw a Venn diagram and use numbers to notate areas of overlap.

  • Jot a chart on your noteboard and solve. This way works best with complicated overlapping set problems.

Practice Set: Overlapping Sets on the GMAT

Answers and explanations at end of chapter

18. Of the 150 employees at company X, 80 are full-time and 100 have worked at company X for at least a year. There are 20 employees at company X who aren’t full-time and haven’t worked at company X for at least a year. How many full-time employees of company X have worked at the company for at least a year?

    • 20

    • 30

    • 50

    • 80

    • 100

19. Three hundred students at College Q study a foreign language. Of these, 110 of those students study French and 170 study Spanish. If at least 90 students who study a foreign language at College Q study neither French nor Spanish, then the number of students who study Spanish but not French could be any number from

    • 10 to 40

    • 40 to 100

    • 60 to 100

    • 60 to 110

    • 70 to 110

20. The Financial News Daily has 25 reporters covering Asia, 20 covering Europe, and 20 covering North America. Four reporters cover Asia and Europe but not North America, 6 reporters cover Asia and North America but not Europe, and 7 reporters cover Europe and North America but not Asia. How many reporters cover all of the 3 continents (Asia, Europe, and North America)?

(1)   The Financial News Daily has 38 reporters in total covering at least 1 of the following continents: Asia, Europe, and North America.

(2)   There are more Financial News Daily reporters covering only Asia than there are Financial News Daily reporters covering only North America.

Answer Key

Practice Set: Averages on the GMAT

1. D

2. E

3. D

4. A

Practice Set: Rates and Speed—Converting Rates on the GMAT

5. C

6. E

7. B

Rates and Speed—Multi-part Journeys on the GMAT

8. E

9. B

10. D

11. C

Practice Set: Combined Rates and Combined Work on the GMAT

12. E

13. C

14. E

Practice Set: Interest Rates on the GMAT

15. C

16. B

17. E

Practice Set: Overlapping Sets on the GMAT

18. C

19. C

20. A

Answers and Explanations

Practice Set: Averages on the GMAT

1.    (D)

Jerry’s average (arithmetic mean) score on the first 3 of 4 tests is 85. If Jerry wants to raise his average by 2 points, what score must he earn on the fourth test?

• 87

• 89

• 90

• 93

• 95

Step 1: Analyze the Question

Jerry has an 85 average after three exams. After his fourth exam, he wants an overall average of 87.

Step 2: State the Task

We need to find the score Jerry must earn on his fourth exam to achieve an overall average of 87.

Step 3: Approach Strategically

The quickest way to solve this problem is to use the “balanced average” approach. To earn an overall average of 87 on four exams, Jerry needs his fourth test to be an 87 plus the number of points that his other three tests were “deficient” with respect to this goal. We know that the average of the first three exams was a score of 85, so each of those three exams, on average, falls short by 87 − 85 = 2 points. Therefore, Jerry needs a score of 87 + 2 + 2 + 2 = 93 on his fourth exam. Choice (D) is correct.

Step 4: Confirm Your Answer

To confirm our answer, we can Backsolve by plugging (D) back into the question: if Jerry scores 85, 85, 85, and 93 on his four exams, that’s an average of . This confirms that (D) is correct.

2.    (E)

The average (arithmetic mean) of all scores on a certain algebra test was 90. If the average of the 8 male students’ grades was 87, and the average of the female students’ grades was 92, how many female students took the test?

• 8

• 9

• 10

• 11

• 12

 

In most questions dealing with averages, you should consider Picking Numbers, setting each term in a group equal to the average of that group. You’ll see in Step 3 how doing so allows you to perform efficient solutions.

Step 1: Analyze the Question

Eight male students take a test, and an unknown number of female students take the test. This question gives us lots of information about averages. Don’t rush to set up algebra yet, as there may be more efficient approaches.

Step 2: State the Task

We have to figure out how many female students took the test. Let’s call that number f.

Step 3: Approach Strategically

This would be a very straightforward Backsolving question, as we could just test an answer choice and see whether it’s consistent with the rest of the information in the question stem. But first, let’s look at a quick math-based approach, taking advantage of the rapid calculations permitted by the “balanced average approach.”

The “balanced average approach” takes advantage of the fact that the sum of the differences between the terms and the average must be zero. Phrased another way, the total amount by which the terms above the average are larger than the average must equal the total amount by which the terms below the average are smaller than the average:

For this problem, here’s how it works: since the only restrictions are that the male students’ scores must average to 87 and that the female students’ scores must average to 92, let’s use Picking Numbers and set all 8 male scores equal to 87 and all female scores equal to 92.

This approach gives us an equation to solve for f.

2f = 24
f = 12

The answer is (E).

Now, let’s try the question using Backsolving. Let’s say that we started with (D). If there were 11 female students whose scores averaged 92, then

The men’s scores are given:

We can set up the formula for the average of the whole group as follows:

But the average is supposed to be 90. We’re too low, so we need more female students’ scores to pull the average up. (E) must be correct.

And here’s the algebraic approach:

The “balanced average approach” almost always permits fewer steps and easier calculations.

Step 4: Confirm Your Answer

Reread the question stem, making sure that you didn’t miss anything about the problem.

3.    (D)

An exam is given in a certain class. The average (arithmetic mean) of the highest score and the lowest score on the exam is equal to x. If the average score for the entire class is equal to y and there are z students in the class, where z > 5, then in terms of x, y, and z, what is the average score for the class, excluding the highest and lowest scores?

• 

• 

• 

• 

• 

Step 1: Analyze the Question

This is a very complicated word problem, with lots of variables, including variables in the answer choices. Whenever there are variables in the answer choices, consider Picking Numbers. When Picking Numbers in a question dealing with averages, pick all the numbers in a group to equal the average of that group.

Step 2: State the Task

Our task is to calculate the average of the class after excluding the highest and lowest scores. Before rushing into the math, let’s think logically about what the task tells us about the correct answer. Since the total size of the class is z, there will be (z − 2) scores after the high score and low scores are eliminated. Since an average is the sum divided by the number of terms, the correct answer must have (z − 2) as a denominator. A quick check of the answer choices reveals that only (C) and (D) might possibly be correct. If you were falling behind on time or unsure of how to approach this question, you could make a 50/50 guess pretty quickly.

Many GMAT word problems have answer choices that can be eliminated without much math.

Step 3: Approach Strategically

Let’s see what we know about the class. The average of the whole class is y, and there are z students in the class. Now what about the high and low scores? We’re told nothing about them, save that their average is x.

This is all pretty abstract, so let’s simplify things by Picking Numbers. Let’s start with the number of students. It must be greater than 5, so let’s say that it’s 6 (z = 6).

Now let’s pick scores for the six students. We could pick all different scores, but the easiest approach of all is to let all six scores be equal to the average of those scores. Let’s say that the class as a whole averages a score of 2. After all, “realism” doesn’t matter when Picking Numbers, only permissibility and manageability. So y = 2.

scores (in order): 2, 2, 2, 2, 2, 2

Now, what are the highest and the lowest? Let’s bold them below for emphasis:

scores (in order): 2, 2, 2, 2, 2, 2

Together, the high and low scores also average 2. We can calculate that so quickly because when all the terms in a group equal the same value, the average is also equal to that value. So x = 2.

Our task is to calculate the average of the class after excluding the high and low scores:

scores (in order): , 2, 2, 2, 2,

Looks like that average will be 2 as well. Now we just need to plug z = 6, y = 2, and x = 2 into the answer choices, looking for the result of 2:

(A)   . Eliminate.

(B)   . Eliminate.

(C)   . Eliminate.

(D)   . Possibly correct.

(E)   . Eliminate.

The answer is (D).

There is an algebraic solution, but you may find this approach more complicated and abstract:

This expression matches (D), which is the correct answer.

Picking Numbers can reduce some of the most complicated average questions to very simple arithmetic.

Step 4: Confirm Your Answer

Reread the question stem, making sure that you didn’t miss anything about the problem.

4.    (A)

If the average (arithmetic mean) of four numbers is 10, how many of the numbers are greater than 10?

(1)   Precisely 2 of the numbers are equal to 10.

(2)   The largest of the 4 numbers is 10 greater than the smallest of the 4 numbers.

Step 1: Analyze the Question Stem

This is a Value question. The introductory information tells us that the average of four numbers is 10. That means the sum of the numbers is 40. From that, we need to determine how many of the numbers are greater than 10. We can make this determination only by knowing how many of the numbers are less than or equal to 10. Thinking critically about the question, we will realize that if three of the numbers are below 10, the remaining number must be above 10. Be careful, though. Knowing that only one number is below 10 would not be sufficient because among the remaining numbers, one or two might be equal to 10.

Step 2: Evaluate the Statements Using 12TEN

Let’s look at Statement (1). If precisely two numbers are equal to 10, then the other two must not be equal to 10. They cannot both be less (or the average would be less than 10), and they cannot both be more, so one must be greater than 10 and one must be less than 10. Statement (1) is sufficient. We can eliminate (B), (C), and (E).

Statement (2) tells us that the largest is 10 greater than the smallest. So the largest must be greater than 10 and the smallest must be less than 10. However, Picking Numbers illustrates that this still leaves us with many possibilities for the middle two numbers. We could have 5, 10, 10, and 15, where one number is greater than 10. We could also have 4, 10, 12, and 14, where two numbers are greater than 10.

Statement (2) is insufficient. Eliminate (D).

There is no need to combine statements in this problem. Choice (A) is correct.

Practice Set: Rates and Speed—Converting Rates on the GMAT

5.    (C)

A truck owner will refuel his vehicle at a rate of 2 gallons of diesel fuel every 5 seconds. If diesel fuel costs $1.25 per gallon, how long will it take for the truck owner to refuel his vehicle with $40 worth of diesel fuel?

• 20 seconds

• 1 minute

• 1 minute 20 seconds

• 20 minutes

• 1 hour 20 minutes

Step 1: Analyze the Question

We are provided with two pieces of information: diesel fuel is $1.25 per gallon and the truck in question refuels at a rate of 2 gallons every 5 seconds. As the choices are in seconds, minutes, and hours, we need to be comfortable converting from one unit to another.

Step 2: State the Task

We are asked to find the amount of time it would take to add $40 of diesel fuel to the truck.

Step 3: Approach Strategically

There’s a lot going on in this problem, so the key is to take things one step at a time. Begin by figuring out the amount of fuel we’re looking to add. At a price of $1.25 per gallon, the truck owner is looking to refuel his vehicle with gallons of fuel. Fuel enters the vehicle at 2 gallons every 5 seconds, so it would take × 5 seconds = 80 seconds to refuel the vehicle. Finally, 80 seconds is 1 minute and 20 seconds, which makes (C) the correct answer.

Step 4: Confirm Your Answer

When evaluating the choices for a question like this, be careful not to read them too quickly, as (E) is designed to look very similar to the correct answer at first glance.

6.    (E)

The moon revolves around the earth at a speed of approximately 1.02 kilometers per second. This approximate speed is how many kilometers per hour?

• 60

• 61.2

• 62.5

• 3,600

• 3,672

Step 1: Analyze the Question

We’re given a rate, and it looks like we’ll have to convert it. We’ll have to pay close attention to the units (seconds, hours, minutes, etc.) so we make the right conversion.

Step 2: State the Task

Our task is to convert 1.02 kilometers per second into terms of kilometers per hour. In other words, we start with this:

And want to end up with this:

Step 3: Approach Strategically

We don’t want to change the “kilometers” term, but we do want to change “seconds” to “hours.” There are 60 seconds per minute and 60 minutes per hour. In other words, the rates are:

Then set up multiplication so that the “seconds” cancel, as do the “minutes”:

So our calculation is 1.02 × 60 × 60. That’s 1.02 × 3,600. While this isn’t the hardest calculation in the world, we don’t have to perform it at all, as only (E) is larger than 3,600.

Step 4: Confirm Your Answer

Reread the question stem, making sure that you didn’t miss anything about the problem. For example, if you solved for kilometers per minute, you’d have selected (B). This step would save you from choosing the wrong answer.

7.    (B)

Magnabulk Corp sells boxes holding d magnets each. The boxes are shipped in crates, each holding b boxes. What is the price charged per magnet, in cents, if Magnabulk charges m dollars for each crate?

• 

• 

• 

• 

• 

Step 1: Analyze the Question

This question gives us a complicated setup, with several different variables. Picking Numbers will probably be a safe approach.

Step 2: State the Task

Our task is to calculate the price per magnet. The word per signals a rate:

Many GMAT word problems have wrong answers that can be eliminated logically, and this is no exception. Since the answer is the amount of money that each magnet costs, we can be sure that the more dollars charged per crate (m), the more money each magnet would cost. In other words, the right answer would have to get bigger as m gets bigger. Answer choices (A), (C), and (E) all have m in the denominator, so those expressions would get smaller as m gets bigger. Those answers can be eliminated.

Step 3: Approach Strategically

We need to solve for “magnets” and “cents.”

What do we know about the number of magnets? Scanning through the question stem, we find “Magnabulk Corp sells boxes holding d magnets.” That means d magnets per box:

What do we know about boxes? “Boxes are shipped in crates, each holding b boxes.” That’s b boxes per crate:

What do we know about the crates? “Magnabulk charges m dollars for each crate.” That’s m dollars per crate:

And, of course, dollars convert to cents at the rate of 100 cents per dollar:

Then set up multiplication such that cents are in the numerator, magnets in the denominator, and everything else cancels:

That’s choice (B).

Picking Numbers will also work well. Just choose permissible and manageable numbers. Let’s say d = 2, b = 5, and m = 4. With these numbers substituted, the question stem reads as follows:

Magnabulk Corp sells boxes of 2 magnets. The boxes are shipped in crates, each holding 5 boxes. What is the price charged per magnet, in cents, if Magnabulk charges $4 for each crate?

Each crate holds 10 magnets, so 400 cents for 10 magnets is 40 cents per magnet. Plugging in d = 2, b = 5, and m = 4:

(A)   . Eliminate.

(B)   . Possibly correct.

(C)   . Eliminate.

(D)   . Eliminate.

(E)   . Eliminate.

Step 4: Confirm Your Answer

Reread the question stem, making sure that you didn’t miss anything about the problem. For example, the problem uses m for dollars and d for magnets instead of vice versa.

Rates and Speed—Multi-part Journeys on the GMAT

8.    (E)

A motorcyclist started riding at highway marker A, drove 120 miles to highway marker B, and then, without pausing, continued to highway marker C, where she stopped. The average speed of the motorcyclist, over the course of the entire trip, was 45 miles per hour. If the ride from marker A to marker B lasted 3 times as many hours as the rest of the ride, and the distance from marker B to marker C was half of the distance from marker A to marker B, what was the average speed, in miles per hour, of the motorcyclist while driving from marker B to marker C?

• 40

• 45

• 50

• 55

• 60

Step 1: Analyze the Question

A motorcyclist rides from highway marker to highway marker for different distances and durations and at different speeds. With so many pieces of information given for this multi-stage journey, we should be prepared to track it all with a table.

Step 2: State the Task

We must find the average speed, in miles per hour, of the motorcyclist while driving from marker B to marker C.

Step 3: Approach Strategically

With so much information to keep track of, the best way to approach this problem is first to organize all of the given concrete information into a table:

	Rate	Time	Distance
marker A to marker B			120 miles
marker B to marker C
Total	45 miles per hour

We’re told that the ride from marker A to marker B took three times as long as the ride from marker B to marker C and that the distance from marker B to marker C is half the distance from marker A to marker B. Using t for the amount of time, in hours, that it took to ride from marker B to marker C, we can now fill the following into our table:

	Rate	Time	Distance
marker A to marker B		3t	120 miles
marker B to marker C		t	60 miles
Total	45 miles per hour	4t	180 miles

It took 4t hours to go 180 miles at 45 miles per hour, so t must equal 1. We can now finish the table:

	Rate	Time	Distance
marker A to marker B	40 miles per hour	3 hours	120 miles
marker B to marker C	60 miles per hour	1 hour	60 miles
Total	45 miles per hour	4 hours	180 miles

We’re looking for the average speed from marker B to marker C, which, according to our table, is 60 miles per hour. The correct answer is (E).

Step 4: Confirm Your Answer

When using a table to organize information, check to ensure that the given pieces of information are filled into the correct parts of the table. Also remember that unlike time and distance, for which the values in the columns sum to get the total, the only way to calculate rate is to divide the distance for that row by the corresponding time.

9.    (B)

A canoeist paddled upstream at 10 meters per minute, turned around, and drifted downstream at 15 meters per minute. If the distance traveled in each direction was the same, and the time spent turning the canoe around was negligible, what was the canoeist’s average speed over the course of the journey, in meters per minute?

• 11.5

• 12

• 12.5

• 13

• 13.5

Step 1: Analyze the Question

A canoeist goes one rate in one direction, turns around, and goes back at a different rate. Whenever you deal with one entity that has different rates at different times, set up a chart to track the data. Otherwise, you’ll find yourself in a six-variable, six-equation system that will take a long time to work through.

Also, notice that although the distance and time are never mentioned, there are no variables in the answer choices. Whenever variables will cancel out, consider Picking Numbers.

Step 2: State the Task

Our task is to calculate her average speed for the whole journey.

Step 3: Approach Strategically

The formula is this:

But we’re seemingly told nothing about time, and the only thing we know about distance is that it is the same in both directions. So what to do? As with almost every problem involving a multi-part journey, set up this chart:

	Rate	Time	Distance
part 1 of trip
part 2 of trip
entire trip

Now plug in the data we’re given:

	Rate	Time	Distance
part 1 of trip	10
part 2 of trip	15
entire trip

Now we see clearly that we’ll be able to know the time if we know something about the distance. Since we know whatever variable we put in place will cancel out by the time we get to the answer choices, let’s just pick a number for distance—one that will fit neatly with a rate of 10 and a rate of 15. A distance of 30 should work well:

	Rate	Time	Distance
part 1 of trip	10		30
part 2 of trip	15		30
entire trip

At this point, we can fill in the rest of the chart very straightforwardly. The entire distance is 60. The time taken upstream must be 3, and the time taken downstream must be 2. That makes the entire time 5.

	Rate	Time	Distance
part 1 of trip	10	3	30
part 2 of trip	15	2	30
entire trip		5	60

The speed for the entire trip, then, is . The answer is (B).

Step 4: Confirm Your Answer

Reread the question stem, making sure that you didn’t miss any information in the problem.

10.   (D)

A truck driver drove for 2 days. On the second day, he drove 3 hours longer and at an average speed of 15 miles per hour faster than he drove on the first day. If he drove a total of 1,020 miles and spent 21 hours driving during the 2 days, what was his average speed on the first day, in miles per hour?

• 25

• 30

• 35

• 40

• 45

Step 1: Analyze the Question

This is another multi-stage journey question. So despite the intimidating presentation, we know that we will transfer the data from the question stem into this chart:

	Rate	Time	Distance
day 1
day 2
entire trip

Step 2: State the Task

We’re solving for speed on the first day, which is the top-left box of the chart.

Step 3: Approach Strategically

Before you get too worried about what your solution will be, plug the data into the chart to help organize your thinking. The first thing we read is “On the second day, he drove 3 hours longer … than he drove on the first day.” We know the total time will be 21 hours, so we can’t just pick a number. Let’s use t for time on the first day. That makes time on the second day t + 3.

Similarly, “On the second day, he drove … at an average speed of 15 miles per hour faster than he drove on the first day,” allows us to say that if r is speed on the first day, then r + 15 is speed on the second day. The rest of the data is simply numerical:

	Rate	Time	Distance
day 1	r	t
day 2	r + 15	t + 3
entire trip		21	1,020

Since the total of the two days’ times will be the time for the entire trip, we can say:

We can put that into the chart:

	Rate	Time	Distance
day 1	r	9
day 2	r + 15	12
entire trip		21	1,020

That allows us to find distance for each day by multiplying (Rate × Time = Distance).

	Rate	Time	Distance
day 1	r	9	9r
day 2	r + 15	12	12(r + 15)
entire trip		21	1,020

Now we have an equation that will allow us to solve for r, which is what we’re looking for—speed on Day 1. Since the total of the two days’ distances will be the distance for the entire trip, we can say:

The answer is (D).

Step 4: Confirm Your Answer

Reread the question stem, making sure that you didn’t miss anything about the problem. For example, if you inverted some information about the two days, you could end up with (A).

11.   (C)

Did Jon complete a journey of 40 kilometers in less time than it took Ann to complete the same journey?

(1)   Jon traveled at an average speed of 30 kilometers per hour for the first 10 kilometers and then at an average speed of 15 kilometers per hour for the rest of the journey.

(2)   Ann traveled at an average speed of 20 kilometers per hour for the entire journey.

Step 1: Analyze the Question Stem

This is a Yes/No question. The stem asks us to compare the time it took two people to complete the same journey of 40 km. Remember that Rate = . Since we’re given the distance in the question stem, we’ll need information adequate to determine both travelers’ rates of speed.

Step 2: Evaluate the Statements Using 12TEN

Statement (1) gives us Jon’s speed. Together with the question stem, this is enough information to determine the time required by Jon to complete the journey. But we get no information about Ann. Statement (1) is insufficient. We can eliminate (A) and (D).

Statement (2) and the question stem together give enough information to determine the time Ann took to complete the journey but nothing about Jon. Statement (2) is also insufficient. We can eliminate (B).

Combining the two statements, we have enough to determine both the time Jon took to complete the journey and the time Ann took to complete the journey. Note that we don’t need to calculate their times. Knowing that we could calculate them is enough. Choice (C) is correct.

Practice Set: Combined Rates and Combined Work on the GMAT

12.   (E)

Kathleen can paint a room in 3 hours, and Anthony can paint an identical room in 5 hours. How many hours would it take Kathleen and Anthony to paint both rooms if they work together at their respective rates?

• 

• 

• 

• 

• 

Step 1: Analyze the Question

Kathleen takes 3 hours to paint a room, and Anthony takes 5 hours to paint the same room.

Step 2: State the Task

The question asks us to find the amount of time it would take Kathleen and Anthony, working together at their respective rates, to paint two rooms.

Step 3: Approach Strategically

We could use the combined work formula to solve this problem, but a bit of Critical Thinking allows us to avoid doing any math at all. Since it takes Kathleen 3 hours to paint a room and it takes Anthony longer than that to paint the same room, it must take the two of them working together longer than 3 hours to paint two rooms. Only (E) is greater than 3, so it must be correct.

Step 4: Confirm Your Answer

Let’s confirm our answer using the combined work formula, , where T is the time Kathleen and Anthony need working together to paint one room, and A and B are the respective amounts of time that it would take each worker individually. Plug the given rates into this formula and solve:

So it takes Kathleen and Anthony hours to paint one of the rooms together. There are two rooms, so it will take them hours to do both. This confirms (E) as the correct answer. Make sure you read the question stem carefully so that you don’t accidentally choose (C), which is the amount of time it would take them to paint only one room together.

13.   (C)

Train A left Centerville Station, heading toward Dale City Station, at 3:00 p.m. Train B left Dale City Station, heading toward Centerville Station, at 3:20 p.m. on the same day. The trains rode on straight tracks that were parallel to each other. If Train A traveled at a constant speed of 30 miles per hour and Train B traveled at a constant speed of 10 miles per hour, and the distance between the Centerville Station and Dale City Station is 90 miles, when did the trains pass each other?

• 4:45 p.m.

• 5:00 p.m.

• 5:20 p.m.

• 5:35 p.m.

• 6:00 p.m.

Step 1: Analyze the Question

In some ways, this problem feels like a multi-stage journey problem (one stage starts at 3:00 p.m. and another at 3:20 p.m.), and in others, it reads like a combined rates problem (two different trains, each moving at once). Proceed with caution but also with confidence that you are using good methodology. Begin as you do for any word problem: make sure you understand the basic situation by paraphrasing it.

Two trains, 90 miles apart, start moving toward each other at different times. One train moves at 30 mph, the other at 10 mph.

Step 2: State the Task

Our task is to determine the times at which the trains pass each other, which is to say when they will be at the same point on these 90-mile tracks.

Step 3: Approach Strategically

There are no variables in the answers to this problem—we get actual values for everything. That often means Backsolving will work. To Backsolve, we’d take a time from the answer choice and see how far each train has moved. The correct answer should put the two trains in the same location.

Let’s use (B) first—where are the trains at 5:00 p.m.? Train A has been traveling 2 hours at 30 mph, so it’s traveled 60 miles, putting it 30 miles away from Dale City. Train B left 20 minutes (or of an hour) later. So it’s only been traveling hours at 10 mph, putting it miles—or 16 miles—away from Dale City. A distance of 13 miles still separates the trains, so they must travel longer before they pass. Eliminate (B) and (A).

Traditionally, you’d want to test (D) at this point. But 35 minutes is an awkward fraction of an hour so it might be faster to work with other choices. At 5:00, the trains were only 13 miles apart, and Train A alone travels 30 miles per hour. In another hour, Train A will have long overshot the target. So (E) seems an unlikely answer. Let’s test (C). If the trains are still too far away, we’ll choose (D).

At 5:20, Train A has been traveling for 2 hours at 30 mph for a total of 70 miles. Train B has been traveling for exactly 2 hours at 10 mph for a total of 20 miles. Together, the trains have gone exactly 90 miles. In other words, Train A is 20 miles away from Dale City, and Train B has traveled 20 miles from Dale City. The trains are at the same place on the track.

(C) is correct.

Had we conceptualized this as a multi-part journey problem, the chart would have looked like this:

	Rate	Time	Distance
A only
A and B
entire trip

The total distance is the 90-mile separation. The time for “A only” is the 20 minutes ( hour) before Train B begins. The rate for “A only” is clearly 30 mph. What about the rate for “A and B”? Since the two trains are each closing the gap, we could say that they are working together to cover the 90 miles. So we can add their rates: 30 + 10 = 40.

	Rate	Time	Distance
A only	30
A and B	40
entire trip			90

Just filling in blank spots on the chart, we can see that the distance for “A only” will be 10. That leaves 80 miles for the “A and B” distance.

	Rate	Time	Distance
A only	30		10
A and B	40		80
entire trip			90

If D = 80 and R = 40, then time must be 2 for “A and B,” making the total time 2 hours.

	Rate	Time	Distance
A only	30		10
A and B	40	2	80
entire trip		2	90

Thus, 2 hours after 3:00 p.m. is 5:20 p.m. The answer is (C).

Step 4: Confirm Your Answer

Reread the question stem, making sure that you didn’t accidentally skip over any important wrinkles. For example, if you had solved for the time Train A arrived at Dale City, you’d have selected (E). This step would allow you to catch that error and, if nothing else, guess from among the other choices.

14.   (E)

Truck X is 13 miles ahead of Truck Y, which is traveling the same direction along the same route as Truck X. If Truck X is traveling at an average speed of 47 miles per hour and Truck Y is traveling at an average speed of 53 miles per hour, how long will it take Truck Y to overtake and drive 5 miles ahead of Truck X?

• 2 hours

• 2 hours 20 minutes

• 2 hours 30 minutes

• 2 hours 45 minutes

• 3 hours

Step 1: Analyze the Question

With two things moving at the same time, this is a combined rate question. The wrinkle is that instead of each truck moving toward the other, one is catching up to the other. How does that affect the situation?

When two things are each moving toward the other, as in question 13, we can add the rates because it’s as if they are each working to close the gap of 90 miles. In this situation, however, Truck Y is working to close the distance, but Truck X is working to extend it. Instead of working together, they are working against each other. So instead of adding the rates, we need to subtract.

In other words, in one hour, Truck X goes 47 miles ahead, but Truck Y catches up 53 miles—that’s the 47 miles Truck X went and 6 more besides. So the net result is that Truck Y gets 53 − 47, or 6 miles, closer to Truck X.

Step 2: State the Task

How long will it take for Truck Y to overtake and drive 5 miles ahead of Truck X?

Step 3: Approach Strategically

If Truck Y starts 13 miles behind Truck X but winds up 5 miles ahead, it gains a total of 13 + 5 = 18 miles relative to Truck X. It’s closing that distance at a speed of 6 mph.

Using Distance = Rate × Time, we get 18 = 6 × Time. So Time = 3 hours.

The answer is (E).

If we didn’t see that we could just subtract the rates, we could have Backsolved, or perhaps we could have jotted down a quick chart tracking their locations at different times to get a sense of when Truck Y would get 5 miles ahead.

Given the answer choices, we could chart 1 hour (just to get a feel for what’s going on), 2 hours (an answer choice), 3 hours (an answer choice), and 2 hours (since we’ll already know the hour-long changes, taking half of that won’t be so bad). We’ll either know the right answer directly or be able to tell which it is.

Time	Truck Y location	Truck X location	Distance apart
0	0	13	Y 13 mi behind
1	53	60	Y 7 mi behind
2	106	107	Y 1 mi behind
2.5
3

At this point, we might well notice that the distance apart changes 6 miles every hour and that another 6-mile gain will put Truck Y in exactly the right spot:

Time	Truck Y location	Truck X location	Distance apart
0	0	13	Y 13 mi behind
1	53	60	Y 7 mi behind
2	106	107	Y 1 mi behind
2.5
3	159	154	Y 5 miles ahead

Choice (E) is correct.

As with many GMAT problems, assessing the situation clearly and logically is the key to finding an efficient solution.

Step 4: Confirm Your Answer

Reread the question stem, making sure that you didn’t accidentally skip over any important aspects of the situation.

Practice Set: Interest Rates on the GMAT

15.   (C)

If John invested $1 at 5 percent interest compounded annually, the total value of the investment, in dollars, at the end of 4 years would be

• (1.5)⁴

• 4(1.5)

• (1.05)⁴

• 1 + (0.05)⁴

• 1 + 4(0.05)

Step 1: Analyze the Question

John is investing $1 at 5 percent interest compounded annually.

Step 2: State the Task

We need to find the expression that represents the value of John’s investment after 4 years.

Step 3: Approach Strategically

The formula for compound interest is Principal × (1 + r)^t, where r is the interest rate per time period expressed as a decimal and t is the number of time periods. So $1 invested at 5 percent interest compounded annually for 4 years would be 1 × (1 + 0.05)⁴ = (1.05)⁴. Choice (C) is correct.

Step 4: Confirm Your Answer

As with many problems that simply ask for translation rather than calculation, we can confirm our answer by realizing why each of the wrong choices is wrong. (A) and (B) both involve an interest rate of 50%, which is much too large. (D) would actually result in less total interest than if John had simply invested for a single year (recall that positive quantities less than 1 raised to a power greater than 1 become smaller), and (E) would be the amount if John had invested at 5 percent simple interest. We have confirmed (C) as the correct answer.

16.   (B)

The amount of an investment will double in approximately 70/p years, where p is the percent interest, compounded annually. If Thelma invests $40,000 in a long-term CD that pays 5 percent interest, compounded annually, what will be the approximate total value of the investment when Thelma is ready to retire 42 years later?

• $280,000

• $320,000

• $360,000

• $450,000

• $540,000

Step 1: Analyze the Question

We’re given a piece of information other than just interest rate and principal. How might it influence our solution? We may not be sure yet, but it’s unlikely to be extraneous information. Also, the word approximate should make us happy: we may be able to estimate the math.

Step 2: State the Task

What is the approximate value of a $40,000 investment that earns 5% compound annual interest in 42 years?

Step 3: Approach Strategically

This problem is a great example of why strategic reading and logical analysis should precede any math. If we rush blindly into math, we’d get Total = $40,000 × (1.05)⁴². Even estimating, there’s no way to solve that accurately without a calculator.

But what if we approached the problem strategically? We’re asked for “total value of the investment.” Scan through the question stem looking for information related to total value, and we find “the amount of an investment will double in approximately years, where p is the percent interest.”

Then we search for the interest and discover that it is 5%. So we now know that the investment will double every years—or every 14 years.

Now we look for how many years the money is invested and see that it’s 42 years: 42 is 14 × 3, so 42 years means 3 doublings. The total value will be approximately $40,000 × 2 × 2 × 2 = $40,000 × 8 = $320,000.

The answer is (B).

Step 4: Confirm Your Answer

Reread the question stem, making sure that you didn’t accidentally skip over any important aspects of the situation.

17.   (E)

A certain account pays 1.5 percent compound interest every 3 months. A person invested an initial amount and did not invest any more money in the account after that. If after exactly 5 years, the amount of money in the account was T dollars, which of the following is an expression for the original number of dollars invested in the account?

• (1.015)⁴T

• (1.015)¹⁵T

• (1.015)²⁰T

• 

• 

Step 1: Analyze the Question

The question addresses compound interest. It’s worth noting that the interest is not annual interest but is compounded every 3 months. Since there are 12 months per year, the interest compounds 4 times per year.

Although there are variables in the answer choices, Picking Numbers probably won’t be needed, as the answers are just various ways that the interest formula might be constructed.

Step 2: State the Task

We are solving for the original amount in the account. Let’s call that P for principal.

Step 3: Approach Strategically

Since the question seems all about whether we can properly construct the compound interest formula, let’s make sure that we carefully set up each part. The general formula is this:

T = P(1 + r)^t

All of the answer choices use 1.015, so we know for certain—if we didn’t already—that inside the parentheses must be 1.015:

T = P(1.015)^t

Since interest is compounded 4 times per year for 5 years, t = 4 × 5 = 20.

The answer is (E).

Step 4: Confirm Your Answer

Reread the question stem, making sure that you didn’t accidentally skip over any important aspects of the situation. For example, if you initially missed the “every 3 months” part of the stimulus, you would have been looking for an answer with an exponent of 4. This step gives you the chance to catch such errors.

Practice Set: Overlapping Sets on the GMAT

18.   (C)

Of the 150 employees at company x, 80 are full-time and 100 have worked at company X for at least a year. There are 20 employees at company X who aren’t full-time and haven’t worked at company X for at least a year. How many full-time employees of company X have worked at the company for at least a year?

• 20

• 30

• 50

• 80

• 100

Step 1: Analyze the Question

This question provides us with a plethora of information about the employees of a company, so we should plan on organizing it all with a table.

Step 2: State the Task

We are looking for the number of full-time employees who have worked at the company for at least a year.

Step 3: Approach Strategically

With so much information to keep track of, the best way to approach this problem is to first organize all of the given concrete information into a table:

	<1 year	≥1 year	Total
part-time	20
full-time			80
Total		100	150

With everything organized clearly, we can now figure out the blanks:

	<1 year	≥1 year	Total
part-time	20	50	70
full-time	30	50	80
Total	50	100	150

We’re looking for the number of full-time employees who have worked at the company for at least a year, which, according to our table, is 50. Choice (C) is correct.

Step 4: Confirm Your Answer

When using a table to organize information, check to ensure that the given pieces of information are filled into the correct parts of the table.

19.   (C)

Three hundred students at College Q study a foreign language. Of these, 110 of those students study French and 170 study Spanish. If at least 90 students who study a foreign language at College Q study neither French nor Spanish, then the number of students who study Spanish but not French could be any number from

• 10 to 40

• 40 to 100

• 60 to 100

• 60 to 110

• 70 to 110

Step 1: Analyze the Question

Since students are split into two potentially overlapping sets—those who study Spanish and those who study French—this is an overlapping sets problem. To help organize the data, draw a table:

	French	not French	Total
Spanish
not Spanish
Total

Step 2: State the Task

We need to find a valid range, with both an upper and lower limit, for those who are in the “Spanish and not French” category.

Step 3: Approach Strategically

We have a piece of data with a lower limit (“at least 90”) but nothing with an upper limit. It’s hard to know intuitively what that upper limit should be, so let’s put in the data from the question stem so we can understand the problem more clearly:

	French	not French	Total
Spanish			170
not Spanish		at least 90
Total	110		300

From this, we can figure out how many do not study each language:

	French	not French	Total
Spanish			170
not Spanish		at least 90	130
Total	110	190	300

Now that we have a clearer understanding of the situation, we can understand more about the limitations on the “at least 90” group. Since the total of the “not Spanish” row is 130, there could be no more than 130 in the “not Spanish and not French” category. (More than 130 would require a negative number of students in the “French and not Spanish” category.)

	French	not French	Total
Spanish			170
not Spanish		90 to 130	130
Total	110	190	300

The “Spanish and not French” category must add with “90 to 130” to yield 190. There could be as many as 100 (since 100 + 90 = 190) or as few as 60 (since 60 + 130 = 190).

The answer is (C).

Step 4: Confirm Your Answer

Reread the question stem, making sure that you didn’t accidentally misread anything. For example, (E) is the value of the “Spanish and French” category.

20.   (A)

The Financial News Daily has 25 reporters covering Asia, 20 covering Europe, and 20 covering North America. Four reporters cover Asia and Europe but not North America, 6 reporters cover Asia and North America but not Europe, and 7 reporters cover Europe and North America but not Asia. How many reporters cover all of the 3 continents (Asia, Europe, and North America)?

(1)   The Financial News Daily has 38 reporters in total covering at least 1 of the following continents: Asia, Europe, and North America.

(2)   There are more Financial News Daily reporters covering only Asia than there are Financial News Daily reporters covering only North America.

Step 1: Analyze the Question Stem

This is a Value question. It is asking for the number of reporters who cover all three continents.

There’s a lot of information, and it begs to be simplified. As the reporters may cover one, two, or three of the continents, this is an overlapping sets question. But with three sets, a chart-based approach would be too unwieldy. The best way to visualize three overlapping sets is with a Venn diagram. We’ll take care to put the totals for each continent just outside the circles so that we don’t lose the distinction between the number covering a continent and the number covering only that continent.

So that we know what we need, let’s put an x in the “all three continents” space on the diagram.

We can continue to analyze our information. A total of 25 reporters cover Asia, 6 of whom also cover only North America, 4 of whom also cover only Europe, and x of whom also cover both Europe and North America.

Therefore, 25 − (6 + 4 + x), or 15 − x, cover only Asia.

Similarly, 20 − (4 + 7 + x), or 9 − x, cover only Europe, and 20 − (7 + 6 + x), or 7 − x, cover only North America.

Let’s put that into our diagram:

Many kinds of information, therefore, would allow us to figure out the number of reporters who cover all three continents. Any data that lets us make an equation with x would be sufficient.

Step 2: Evaluate the Statements Using 12TEN

Statement (1) says that the grand total of all the reporters is 38. In other words, if we added up all the various subcategories, we’d get 38. That’s sufficient, as the only variable in that equation would be x. Eliminate (B), (C), and (E).

Although you wouldn’t want to set the whole thing up (you’d stop as soon as you knew that you could set up the equation), here’s what it would be:

(15 − x) + (9 − x) + (7 − x) + 4 + 7 + 6 + x = 38

Once the information from the question stem has been fully simplified and analyzed, we see that Statement (2) is only saying that (15 − x) > (7 − x). That’s just 15 > 7, which is hardly new information. Statement (2) adds nothing and must be insufficient. Choice (A) is correct.

GMAT BY THE NUMBERS: MATH FORMULAS

Now that you’ve learned how to approach math formulas questions on the GMAT, let’s add one more dimension to your understanding of how they work.

Take a moment to try the following question. The next page features performance data from thousands of people who have studied with Kaplan over the decades. Through analyzing this data, we will show you how to approach questions like this one most effectively and how to avoid similarly tempting wrong answer choice types on Test Day.

In an election, candidate Smith won 52 percent of the total vote in Counties A and B. He won 61 percent of the vote in County A. If the ratio of people who voted in County A to County B is 3:1, what percent of the vote did candidate Smith win in County B?

• 25%

• 27%

• 34%

• 43%

• 49%

Explanation

QUESTION STATISTICS

57% of test takers choose (A)

16% of test takers choose (B)

10% of test takers choose (C)

12% of test takers choose (D)

5% of test takers choose (E)

Sample size = 5,221

The question statistics above show that this is a hard question, as many test takers get it wrong. But no single wrong answer stands out as a common trap. This is the sign of a problem that is challenging because most test takers simply don’t know how to approach it; they aren’t seduced by particular traps—they just can’t find the answer. In fact, after reading the problem, you may have been surprised that even 57% of test takers got this one right! Two factors contribute to this seemingly high percentage: first, the percentages of test takers who chose the other choices suggest that somewhere between 10% and 15% may have just made a lucky guess, bringing the percentage who actually solved it correctly down to perhaps 40–45%. Second, these statistics are drawn from students in a Kaplan course. This question fits a common pattern in GMAT problems, and we teach a formula that handles it well.

The pattern we’re referring to is a problem in which two entities, each with a different proportion or average, are combined to form a total with an overall proportion or average that’s different from its individual components. Here, it’s County A and County B that combine to give Smith 52% of the total vote. Here’s the formula that handles this problem:

(Proportion A)(Weight of A) + (Proportion B)(Weight of B) = (Total proportion)(Total weight)

Here you must solve for the proportion of the vote Smith won in County B, which you can represent with the variable b. You can now just plug the data into the formula and solve:

Answer choice (A) is correct.

By learning these classic formulas and problem solving techniques, you’ll find yourself comfortably handling problems that stump most other test takers.

More GMAT by the Numbers …

To see more questions with answer choice statistics, be sure to review the full-length CATs in your online resources.