Mastering GMAT Rate Problems

Rate problems are a staple of the GMAT’s quantitative section, often involving scenarios that test your ability to calculate speeds, distances, and times. 

These problems require not just mathematical skills but also the ability to think critically and logically. The quantitative section of the GMAT works by using simple mathematical relationships to create challenging problems. This makes rate problems – which depend on simple relationships that are surprisingly versatile in their presentation and arrangement – a favorite and frequent category for GMAT quant. 

While these problems may be a favorite for the writers of the test, they are rarely a favorite for test-takers! 

With a solid grasp of the fundamentals, the right mathematical tools, and some practice, you’ll be able to solve these problems faster than you can say “respective constant rates.” 

Key Concepts and Formulas

Before diving into problem-solving strategies, let’s review some essential concepts and formulas:

1. Rate Formula:

Rate = Work (or Distance) / Time

This formula can be rearranged depending on the given variables:

Work (or Distance) = Rate x Time

Time = Work (or Distance) / Rate

2. Average Rate

When dealing with multiple segments of a journey, the average rate is calculated using the total distance and total time:

Average Rate = Total Distance/Total Time

3. Relative Rates

When two objects are moving towards or away from each other, their relative rate is the sum or difference of their individual rates.

This is the same relationship and the same equation expressed in terms of each of the three variables. The first form, rate = work/time, is the most fundamental starting point because quite simply, rate is work/time. There is no single unit for expressing a rate. Instead, we have to use both a work unit and a time unit, connected by a word laden with mathematical meaning: per.

Elijah Mize (Apex GMAT Instructor) – “As an exercise in unit conversions, my high school physics teacher made us express the speed of light in furlongs per fortnight! A furlong is ⅛ of a mile, and a fortnight is two weeks or fourteen days. Of course, our starting point was the speed of light expressed in meters per second. But whether you use meters per second, furlongs per fortnight, or marathons per leap year, any expression of speed is “distance unit per time unit,” with the word “per” acting like a fraction bar.”

This brings up an important point about rate problems: problems involving machines completing jobs or producing widgets are fundamentally the same as problems involving speed, distance, and time! If one type of problem makes more sense to you than the other, you can take advantage of this to help your comprehension of the other type of problem.

If you like machines, you can think of cars/trains/bikes/buses as machines that “produce” kilometers or miles, where the “work” done by a moving body is the distance it covers in a given time.

If you understand speed, distance, and time, you can remember that rate, work, and time are mathematically related in the exact same way. To say it another way, “rate” is the general term for something happening in a given amount of time, and  “speed” is simply what we call a rate when the thing happening is motion (which needs distance units to be expressed). 

It is not enough to know these simple formulas by rote: you need to understand the network of direct and inverse relationships between the variables. When a single variable is isolated, or solved for, or expressed in terms of the others (three ways of saying the same thing), it is directly related to the variables on the same side of any fraction bars and inversely related to the variables on the opposite side of any fraction bars.

Remember that if an isolated variable is not in a fraction at all, as in the left sides of the formulas above, it can be thought of as the numerator of a fraction with a denominator of 1. In other words, if it’s not a denominator, it’s a numerator.

Direct and inverse relationships exist between the variables

Based on this system, the following direct and inverse relationships exist between the variables:

• Rate and Time (or Speed and Time) are inversely related.
• Work and Time (or Distance and Time) are directly related.
• Work and Rate (or Distance and Speed) are directly related.

A direct relationship means that the values of variables increase together and decrease together. An inverse relationship means that the values of variables change in opposite directions: if the value of one variable increases, the value of the other decreases, and if the value of one variable decreases, the value of the other increases. Let’s apply this to the variables in the formula.

• As the rate increases, the amount of work done in a given time increases.
• As the rate increases, the amount of time to do a given amount of work decreases.
• As time increases, the amount of work done at a given rate increases.
• As time increases, the rate to complete a given amount of work decreases.
• As work increases, the rate to complete the job in a given amount of time increases.
• As work increases, the time to complete the job at a given rate increases.

Try “translating” these relationships into speed, distance, and time to help solidify the concepts.

Since the rate = work/time formula is “pure variables” and involves no exponents, coefficients, or constants, the change in any variable given the change in another is remarkably predictable and consistent. If the value of one variable changes by a given factor, the value of a directly related variable changes by the same factor, and the value of an inversely related variable is multiplied by the reciprocal of that factor.

For example, if a car’s average speed over a given distance increases from 60mph to 75mph (multiplication of the value by a factor of 5/4), the time it takes to cover that distance becomes 4/5 (the reciprocal of 5/4) of what it was, because speed and time are inversely related.

If the time in which a machine needs to complete a job is halved, the rate at which the machine must work to finish in time doubles. Or if the time allowed to complete the job is multiplied by a factor of 3/2 (like an increase from 2 hours to 3 hours or from 2 days to 3 days), the machine may work at ⅔ of its prior rate. If the size of a job is multiplied by a factor of 5/3, a machine (or person) must work 5/3 as fast in order to finish in the same amount of time because work and rate are directly related. If the machine or person simply keeps working at the same rate, the job will take 5/3 as long as it would have taken, because work and time are also directly related.

Hopefully these relationships are starting to make sense. Let’s try them out on some official GMAT quant problems:

Official GMAT quant problems

Carl averaged 2m miles per hour on a trip that took him h hours. If Ruth made the same trip in ⅔h hours, what was her average speed in miles per hour?

(A) (1/3)mh

(B) (2/3)mh

(C) m

(D) (3/2)m

(E) 3m

This one follows the form of the examples above. If Ruth completes the trip in ⅔ the time it took Carl, she must have been traveling 3/2 times as fast as Carl, on average, because time and rate (or speed) are inversely related. Carl’s speed was 2m, so Ruth’s speed was 3/2 * 2m = 3m. The correct answer is E. The inverse relationship of speed and time is all you need for this problem.

Strategies for Solving Rate Problems

1. Read the Problem Carefully: Understand what is being asked. Identify the known quantities (distance, rate, time) and what needs to be found.
2. Set Up the Equation: Use the rate formula and plug in the known values.
3. Solve for the Unknown: Rearrange the equation to solve for the unknown variable.
4. Check Units: Ensure that the units of distance, rate, and time are consistent. Convert units if necessary.

Example Problem

Problem: A car travels 150 miles at a constant speed of 50 miles per hour. How long does the trip take?

Solution:

1. Identify the known values: Distance = 150 miles, Rate = 50 miles per hour.
2. Set up the equation:

Time = Work (or Distance) / Rate = 150 miles/50 miles per hour

1. Solve for time:

Time = 3 hours

Common Types of Rate Problems

Uniform Motion Problems

These involve objects moving at a constant speed. The key to solving these is understanding the direct relationship between distance, rate, and time.

Example: A train travels at 60 miles per hour for 3 hours. How far does it travel?

Solution:

Distance=Rate×Time=60 miles per hour×3 hours=180 miles

Distance=Rate×Time=60 miles per hour×3 hours=180 miles

Objects Moving in the Same Direction

When two objects move in the same direction, they are effectively working against each other in terms of changing the distance between them. Imagine two cars on a highway: if Car A is traveling at 60 mph and Car B is traveling at 50 mph, Car A is moving away from Car B at a rate of 10 mph. The relative speed (difference in their speeds) determines how the distance between them changes over time.

Objects Moving in Opposite Directions

When two objects move in opposite directions, they work together to change the distance between them. For instance, if two trains start from the same point and travel in opposite directions—one north at 60 mph and the other south at 40 mph—the distance between them increases by the sum of their speeds, or 100 mph.

Problems Involving Two Moving Objects

These can include objects moving towards each other, away from each other, or in the same direction.

Example: Two cyclists start from the same point and ride in opposite directions. One rides at 10 miles per hour and the other at 15 miles per hour. How far apart will they be after 2 hours?

Solution:

Total Distance = (Rate1 +Rate2 ) × Time = (10 mph+15 mph) × 2 hours = 50 miles

When two objects move, their work is defined by the change in distance between them. This concept can be counterintuitive: if two objects are moving in the same direction, they work against each other. For example, imagine one pump is filling a pool while another is emptying it. Any distance one object covers is partially canceled out by the other object’s movement. So, a getaway car increases the distance from pursuing police vehicles, while the police work to reduce this distance.

In contrast, when objects move in opposite directions—either towards or away from each other—they work together to change the distance between them, either decreasing or increasing it.

Regardless of whether objects move in the same or opposite directions, their positions must be considered. For instance, if two trains start 100 miles apart, travel towards each other until they meet, and then continue moving away to end up 30 miles apart, the total work done is 130 miles (100 miles to meet plus 30 miles of separation), not 70 miles (100 miles minus 30 miles). The trains collectively cover 100 miles to meet and another 30 miles to separate.

Tips and Tricks

1. Break Down Complex Problems: If the problem involves multiple segments or stages, break it down into simpler parts and solve each part step by step.
2. Use Diagrams: Drawing a diagram can help visualize the problem, especially for problems involving relative motion.
3. Practice Mental Math: Being quick with basic arithmetic can save time and reduce errors.
4. Double-Check Your Work: Always review your calculations to ensure accuracy, especially under timed conditions.

Common Pitfalls

1. Ignoring Units: Ensure all units are consistent throughout the problem.
2. Misinterpreting the Problem: Read the problem carefully to understand what is being asked.
3. Overlooking Relative Motion: Pay attention to whether objects are moving towards each other or away from each other.

Advanced Example: Challenging Work Rate Problem

If a certain machine produces bolts at a constant rate, how many seconds will it take the machine to produce 300 bolts?

1. It takes the machine 56 seconds to produce 40 bolts.
2. It takes the machine 1.4 seconds to produce 1 bolt.

(A) Statement (1) alone is sufficient, but statement (2) alone is not sufficient.

(B) Statement (2) alone is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are not sufficient.

The question asks “how many seconds,” so the variable we’re solving for is time to complete a given amount of work: 300 bolts. With the value for work filled in, the only remaining variables are time and rate. Therefore if a statement allows us to calculate a rate, time will be the only variable left, and that statement will be sufficient.

Each statement provides us with the amount of time it takes for the machine to complete a given amount of work: 56 seconds for 40 bolts or 1.4 seconds per bolt. Since rates don’t have their own units and can only be expressed as a given amount of work in a given amount of time, these statements are giving us the rate at which the machine produces bolts! We don’t have to calculate anything or actually find how many seconds this machine takes to product 300 bolts. 

Each statement provides the rate at which the machine will work to complete the 300-bolt job, so each statement is sufficient to determine the time in which the machine will complete this job. 

The correct answer is D.

Practice Problems

1. A boat travels 60 miles downstream in 3 hours and returns upstream in 4 hours. What is the speed of the current?
2. Two cars start from the same point. Car A travels north at 40 miles per hour, and Car B travels east at 30 miles per hour. How far apart are they after 2 hours?
3. A cyclist travels for 5 hours. For the first 3 hours, he rides at 10 miles per hour, and for the next 2 hours, he rides at 15 miles per hour. What is his average speed for the entire trip?

By mastering these concepts and strategies, you’ll be well-equipped to tackle GMAT rate problems with confidence. Practice regularly and review your mistakes to continuously improve your problem-solving skills. 

Additional Tips for Mastering GMAT Rate Problems

• Practice Regularly

Regular practice is essential to mastering GMAT rate problems. Use a variety of practice problems to familiarize yourself with different scenarios and question types. This will help you develop the ability to quickly identify the best approach to each problem.

• Use Official GMAT Materials

Official GMAT materials provide problems that are representative of the actual exam. They help you understand the level of difficulty and the types of problems you will encounter. Incorporate these materials into your study plan to ensure you are well-prepared.

• Time Management

Effective GMAT time management is crucial for success. Practice solving rate problems under timed conditions to get used to the pressure of the exam. Aim to solve problems quickly and accurately to ensure you have enough time for all sections.

• Focus on Understanding, Not Memorization

While it’s important to know formulas, understanding the underlying concepts is key. This deeper understanding allows you to adapt to any problem the GMAT throws at you. Focus on why and how formulas work rather than just memorizing them.

• Seek Help When Needed

If you find certain types of rate problems particularly challenging, consider seeking help from a tutor or joining a study group. Explaining problems to others and hearing their perspectives can provide new insights and enhance your understanding.

• Stay Calm and Positive

Finally, maintain a positive mindset. Rate problems can be tricky, but with consistent practice and the right strategies, you can master them. Stay calm during your study sessions and on test day, and remember that each problem is an opportunity to demonstrate your skills.

If you are looking for extra help in preparing for the GMAT, we offer extensive one-on-one GMAT tutoring. You can schedule a complimentary 30-minute consultation call with one of our tutors to learn more!