GMAT Rate Problems: How to Conquer All 3 Types

stopwatch-259375__340Two cars bound for the same GMAT test center leave their houses at the exact same time. Car A travels 30 miles per hour; Car B travels 20 miles per hour. Car A has to travel 20 miles. Car B has to travel 15 miles. Which car will arrive first?

If you couldn’t already tell, we’re going to focus on rate problems in this article!

There are a number of different types of rate problems that you’ll see on the GMAT. (Don’t worry – they’ll all make much more sense than the silly examples I provided above.) In this article, I’ll explain what rate problems are, give you the formulas and equations you need to solve the three most common kinds of rate problems on the GMAT, and give you tips on how to practice for GMAT rate problems.

By the end of this article, you’ll have everything you need to start tackling your GMAT questions at a faster and more accurate rate!

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What Are GMAT Rate Questions?

There are three basic types of rate problems that you’ll see on the GMAT: questions that ask about the rate that work is completed, questions that ask you about the rate of distance traveled, and questions that ask you about interest rates.

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A typical work rate question may tell you about two different people that are working at a particular rate. The question may ask you how long it’ll take for the two people to finish a job as they’re both working. For instance, a question may ask how long it takes two students, working together at different rates, to generate a certain product.

A typical distance rate question may tell you that a car travels along a road at a certain speed for one length of time, then continues traveling at a different speed for another length of time. The question may ask you to determine the average rate of travel for the entire trip. For instance, the question may ask you to determine how long it’ll take a car to get from Point A to Point B, if it’s traveling at two different constant rates of speed.

A typical interest rate question may ask you how much interest you’ll earn per year if you’re earning interest at a set rate for a certain amount of money. For instance, the question may ask you to determine how much money Person A earns in one year if they invest $x$ amount of money at $y$ interest rate.

You’ll see all three types of questions throughout the quantitative section of the GMAT. You’ll see rate questions for both data sufficiency and problem solving questions.

In the next section, I’ll talk about the formulas and equations you need to know for each type of rate problem and give you sample questions for each as well.

 

GMAT Work Rate Problems

As I mentioned before, GMAT work rate problems ask you about the rate of completion of different jobs. While these questions seem pretty simple, GMAT rate problems often trip up test takers.

The basic equation you need to know for GMAT work rate problems is:

$$\amount = \rate*\time$$

What does that mean? Basically, it means that the amount of goods produced is equal to the rate at which the goods are produced multiplied by the time spent producing the goods.

So, for instance, if I were to produce Grumpy Cat stuffed animals at a rate of 2 animals per hour, and I spent two hours producing stuffed animals, my equation would look like this:

$$\amount = 2 \animals/\hour * 2 \hours$$

I create four stuffed animals in two hours.

Obviously, GMAT work rate problems are more complicated than that example. It’s important to be comfortable with the basic equation $amount = rate*time$, though, as well as understand its implications. For instance, if the rate at which I produce stuffed animals doubles, but the time remains the same, the amount of stuffed animals I produce also doubles.

Let’s take a look at using this formula in action in a couple sample questions.

 

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PROBLEM SOLVING WORK RATE SAMPLE QUESTION

Three printing presses, $R$, $S$, and $T$, working together at their respective constant rates, can do a certain printing job in 4 hours. $S$ and $T$, working together at their respective constant rates, can do the same job in 5 hours. How many hours would it take $R$, working alone at its constant rate, to do the same job?

  1. 8
  2. 10
  3. 12
  4. 15
  5. 20

Let’s start by assigning variables for the portion of the job that each machine completes alone. $R$ = the rate at which $R$ makes products; $S$ = the rate at which $S$ makes products; $T$ = the rate at which $T$ makes products.

Now, let’s plug those variables into an equation. We know that $R$, $S$, and $T$ working together for 4 hours make one product. We know that $S$ and $T$ working together for 5 hours make one product. That knowledge yields us these equations:

  • $4R+4S+4T=1$
  • $5S+5T=1$

We’ll isolate the variables from constants in each of these equations, meaning that we end up with:

  • $R+S+T = 1/4$
  • $S+T = 1/5$

Now, we can substitute $1/5$ for $S + T$ in our original equation. That yields us the equation $R + 1/5 = 1/4$. Let’s isolate $R$, then: $R = (1/4) – (1/5)$.

$R$, therefore, equals $1/20$.

Remember, that $1/20$ is in the form of ($amount$/$time$). 1, therefore is the completed job, and 20 is the time it takes to complete the job.

The answer, then, is $E$. It takes $R$ 20 hours to complete the job.


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DATA SUFFICIENCY WORK RATE SAMPLE QUESTION

Three machines, $K$, $M$, and $P$, working simultaneously and independently at their respective constant rates, can complete a certain task in 24 minutes. How long does it take Machine $K$, working alone at its constant rate, to complete the task?

  1. Machines $M$ and $P$, working simultaneously and independently at their respective constant rates, can complete the task in 36 minutes.
  2. Machines $K$ and $P$, working simultaneously and independently at their respective constants rates, can complete the task in 48 minutes.
  1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
  3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
  4. EACH statement ALONE is sufficient.
  5. Statements (1) and (2) TOGETHER are NOT sufficient.

Let’s start by defining what we know. Let’s say that $K$, $M$, and $P$ are the number of minutes it takes machines $K$, $M$, and $P$, respectively, to complete the task. The amount of the task that $K$ can do in one minute is then $1/K$; the amount $M$ can do in one minute is $1/M$; the amount P can do is $1/P$.

If it takes all three machines working together 24 minutes to do the task, they can complete $1/24$ of the task, together, in one minute.

So, it holds that $1/K + 1/M + 1/P = 1/24$.

Now, let’s look at each statement by itself. Remember, for data sufficiency questions, we always want to assess each statement alone first.

Statement (1) says that $M$ and $P$ together can complete the task in 36 minutes. That means that they can complete $1/36$ of the task in 1 minute. In other words, $1/M + 1/P = 1/36$. We can use that information to then solve for a unique value of K by plugging into our original equation:

$1/K + (1/36) = 1/24$. (Note: you don’t actually need to solve data sufficiency questions; you just need to know that you can. I’m showing the completed equation here to show that it can be solved.)

Statement (1), therefore, is sufficient to solve for $K$.

Now let’s forget everything we learned in statement (1) and look at statement (2). Statement (2) tells us that $K$ and $P$ can do the task together in 48 minutes. So $1/K + 1/P = 1/48$, or they can do $1/48$ of the task in one minute.

However, we don’t have any information here to figure out the value of $K$. Since we have no value for $P$, we can have a number of values for $K$. We can’t solve the equation using this information.

Therefore, statement (2) is not sufficient.

The correct answer, then is $A$.

 

GMAT Distance Rate Problems

GMAT distance rate questions often ask you to determine how far someone or something is traveling at a certain speed. You may be asked to calculate average speed or average distance or the time spent traveling. There’s one basic equation that you can use to complete these calculations.

The basic equation for distance is as follows:

$$\distance = \rate*\time$$

 

PROBLEM SOLVING DISTANCE RATE SAMPLE QUESTION

After driving to a riverfront parking lot, Bob plans to run south along the river, turn around, and return to the parking lot, running north along the same path. After running 3.25 miles south, he decides to run for only 50 minutes more. If Bob runs at a constant rate of 8 minutes per mile, how many miles farther south can he run and still be able to return to the parking lot in 50 minutes?

  1. 1.5
  2. 2.25
  3. 3.0
  4. 3.25
  5. 4.75

Let’s start with what we know. We know that so far, Bob has been running for a distance of 3.25 miles. We also know that he is running at a constant rate of 8 minutes per mile. Normally, we express speed in miles per hours, with distance over rate, so this constant rate is expressed differently than the normal, We can use this knowledge to figure out how long Bob has been running for by plugging those numbers into our distance equation.


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$(3.25 miles)*(8 minutes$/$mile) = 26 minutes$. Bob has been running for 26 minutes.

Let’s then say that $X$ is the number of additional minutes Bob can run south before turning around. The number of minutes that he’ll be running north, then, will be $X + 26$.

Since Bob will be running a total of 50 minutes after his initial 26 minutes of running, we can say that his remaining total time running (50) is equal to the rest of his time running south ($X$) and his time running north ($X + 26$). This knowledge gives us the equation: $X + (X + 26) = 50$.

That means that $X = 12$, or, Bob can run for 12 more minutes before turning around. his total remaining time running (50) is equal to the rest of his time running south (x) + his time running north (x+26)

Now, we can plug 12 minutes into our distance rate equation to figure out how many more miles Bob can run.

$(12 minutes)$/$(8 minutes$/$mile$) = $miles$ Bob can run before turning around.

So, Bob can run 1.5 more miles before turning around. The correct answer is A.

 

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DATA SUFFICIENCY DISTANCE RATE SAMPLE QUESTION

On a certain nonstop trip, Marta averaged $X$ miles per hour for 2 hours and $Y$ miles per hour for the remaining 3 hours. What was her average speed, in miles per hour, for the entire trip?

  1. $2x + 3y = 280$
  2. $y = x + 10$
  1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
  3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
  4. EACH statement ALONE is sufficient.
  5. Statements (1) and (2) TOGETHER are NOT sufficient.

For data sufficiency questions, we always evaluate each statement alone first. Let’s look at statement (1) first.

Statement (1) tells us that $2x + 3y = 280$. We know that $X$ and $Y$ are our variables for distance. We also know that Marta traveled for a total of 5 hours.

Marta’s average speed, then is $(2X + 3Y/5)$ miles per hour, as we know that she’s traveling for $X$ miles per hour for 2 hours and $Y miles$/$per hour$ 3 hours and dividing by 5 will help us, then, find the average speed over the entire length of the trip (5 hours).

Since $(2X + 3Y) = 280$, it also follows that $(2X + 3Y)/5 = 280/5$. Statement (1) is sufficient.

Now let’s look at statement (2). Statement (2) tells us that $Y = X + 10$. Let’s plug that back into our initial equation for speed.

$(2X + 3Y)/5 = R$, where $R$ equals speed. Let’s plug in our equation for $Y$.

$[2X + 3(X +10)]/5 = R$, or $X + 6 = R$. However, we don’t have enough information to solve for speed.

Therefore, statement (2) is not sufficient.

The correct answer, then, is A.

 

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GMAT Interest Rate Problems

GMAT interest rate problems typically ask you to determine how much money a person earns if they place a certain amount of money into account at a particular interest rate.

There are a number of formulas that you should memorize to master these questions.

The simple formula for interest is:

Interest = $P x R x T$

  • $P$ = starting principle (or, amount in the account)
  • $R$ = annual interest rate (usually expressed as a percent)
  • $T$ = number of years

GMAT interest rate questions typically ask about more complicated interest questions, as well as simple interest questions. Here are some other formulas you should know for GMAT interest rate questions:

The formula for annual compound interest is:

Annual Compound Interest = $P(1+R)^T$

  • $P$ = starting principle
  • $R$ = annual interest rate
  • $T$ = number of years

The formula for compound interest is:

Compound Interest = $P(1 + R/X)^xt$

  • $P$ = starting principle
  • $R$ = annual interest rate
  • $T$ = number of years
  • $X$ = numbers of times the interest compounds over the year

 

PROBLEM SOLVING INTEREST RATE SAMPLE QUESTION

John deposited $10,000 to open a new savings account that earned 4 percent annual interest, compounded quarterly. If there were no other transactions in the account, what was the amount of money in John’s account 6 months after the account was opened?

  1. $10,100
  2. $10,101
  3. $10,200
  4. $10,201
  5. $10,400

Let’s start by understanding what the question tells us. It tells us that John receives his interest compounded quarterly. That means he receives his one quarter (1/4) of his compound interest every three months. Since John’s interest rate is 4 percent, that means he earns 1 percent of his interest every three months.

Every three months, 1/4 of the compound interest is added to John’s account. This new total then accrues interest for the next quarter.

The question asks us how much money is in John’s account after six months. Six months means that John’s account has accrued interest twice.

The amount of money in John’s account will then be found with the equation: $amount = ($10,000)(1.01)(1.01)$.

So, John will have $10,201 in his account after six months. Lucky John!

The correct answer is D.

 

DATA SUFFICIENCY INTEREST RATE SAMPLE QUESTION

On a certain date, Hannah invested $5,000 at $X$ percent simple annual interest and a different amount at $Y$ percent simple annual interest. What amount did Hannah invest at $Y$ percent simple annual interest?

  1. The total amount of interest earned by Hannah’s two investments in one year was $900.
  2. Hannah invested the $5,000 at 6 percent annual interest.
  1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
  3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
  4. EACH statement ALONE is sufficient.
  5. Statements (1) and (2) TOGETHER are NOT sufficient.

For data sufficiency questions, we always want to assess each statement by itself first.

First, let’s say that $H$ = the amount of money Hannah invested at $Y$ percent. We can use that variable for all of our equations.

We can express Hannah’s earned interest rates with two equations. First, $(X/100)($5,000) = the amount of interest$ Hannah earned from her first account. $(Y/100)(H)$ = $the amount of interest$ Hannah earned from her second account.

Statement (1) says that Hannah earned $900 from both accounts. So, $([X/100][$5,000] + [Y/100][A]) = $900$.

We have no way of further isolating the variables however, so we don’t have enough information to solve for $H$. Statement (1) is not sufficient.

Let’s look at statement (2). Statement (2) tells us that $X = 6$, because the statement tells us that the annual interest rate is 6 percent. However, telling us that $X$ is 6 still doesn’t give us enough information to solve for A. Statement (2) is also not sufficient.

Statements (1) and (2) together are also not sufficient because there is more than one possible value for $H$. The correct answer, therefore, is E: both statements together are not sufficient.

 

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How to Practice for GMAT Rate Problems

While GMAT rate problems may seem tricky, practicing them will significantly increase your ability to solve them on test day. Keep in mind these tips as you’re studying to boost your performance.

 

#1: Memorize the Formulas and Equations

The best way to feel confident when tackling GMAT rate questions is to know your GMAT rate formulas and equations backwards and forwards… literally! For instance, memorize that $distance = rate*time$ and $time = distance/rate$. Spend time memorizing the other equations and formulas that I suggested in the earlier sections of this article. If you spend your time memorizing equations and formulas, you’ll be able to easily understand what a particular question is asking you and what to do with the different variables in the question.

A great way to memorize formulas is by using flashcards. To learn more about the best ways to use flashcards, check out our total guide to the best GMAT flashcards.

 

#2: Remember That Rates Are Like Ratios

While rates may seem pretty complicated, they’re really actually ratios! Or, in another form, fractions. For instantly, fuel efficiency (miles per gallon) is really expressed as a fraction: $\miles/\gallon$. Thinking about rates in terms of fractions with numerators and denominators may help them seem easier to understand.

Remember, because rates are really ratios, we can solve GMAT rate questions by thinking about proportions, just like we would do with normal fractions and ratios. This tip can help if you’re struggling with how to set up your rate equations. Setting up rates as a ratio/fraction will help you think about how the different variables relate to each other.

 

#3: Add or Subtract the Rates

Most GMAT work rate problems will ask you to analyze the work rate of two or more machines or people. For these questions, you’ll likely have to compare individual production with their combined production.

When you’re tackling these problems, remember this fact: you can add or subtract the rates on these questions. You can’t add or subtract times. So, when you’re looking to find a combined amount of goods produced, make sure you’re combining the rates, not the times. For instance, if two machines are creating a certain amount of goods in a set time, you can add or subtract the amount of goods, but the time it takes to make the goods will remain the same.

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#4: Use High-Quality Practice Materials

The best way to prepare for the GMAT is by using real GMAT practice questions in your prep. Real GMAT practice questions will simulate the GMAT’s style and content. Using resources like GMATPrep or the GMAT Official Guide will give you access to real, retired GMAT questions.

As you might’ve noticed from our practice questions, you’ll rarely see a straightforward question on the GMAT that just asks you to use your rate problems skills. You’ll likely have to combine your knowledge of rate problems with your knowledge of arithmetic or number properties or ratios… or all of the above! Practicing GMAT-style questions (real, retired GMAT questions if you can get them) will give you practice at using multiple skills in one question.

 

Review: Solving GMAT Rate Problems

There are three basic types of GMAT rate problems: questions that ask about the rate of work completion, questions that ask about interest rate, and questions that ask about distance.

GMAT rate questions are very formula driven. Spend some time memorizing the equations and formulas in this article to be able to quickly and easily solve any rate problem you encounter on exam day.

 

What’s Next?

Want to set up a steady rate of study for the GMAT? (I promise, the rate puns are over now.) Our detailed GMAT study plan will help you set up an optimal study schedule and give you a solid foundation for improving your GMAT score.

Looking to master another part of the GMAT quantitative section? We have guides to help you build your knowledge in a number of different content areas. Check out our total guides to GMAT geometry and GMAT probability for content-specific guides.

If you’re looking for a more in-depth overview of the quant section, we have our total guide to GMAT quant, as well as a review of the best GMAT quant practice. We also have a complete guide to mastering the three question types on the GMAT verbal section, if you’d like to focus on a different area of the test.

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