What’s on the GRE Work Rate Problems?

Acing the GRE Quantitative Reasoning section requires solving various challenging questions, and rate problems are some of the most common yet difficult. These word problems test your ability to apply mathematical concepts like distance, time, and speed to real-world scenarios. However, mastering rate problems can be incredibly rewarding and significantly boost your GRE score.

This blog digs deep into the world of GRE work rate problems. We’ll explore the two main types you’ll come across on the exam: rate-time-distance and rate-time-work questions. We’ll break down how these problems are presented and provide you with strategies to solve them confidently.

What is a Rate?

In math, a rate describes how something changes in relation to another thing, specifically how much of one quantity occurs per unit of another quantity. It’s essentially a measurement of speed or efficiency.

Imagine you’re baking cookies. A rate would tell you how many cookies you can bake in a certain amount of time, for instance, 10 cookies per hour. Here, cookies are one quantity, and time (hours) is the other. The “per” indicates the relationship between them: 10 cookies for every 1 hour.

Rates are most helpful when the two quantities being compared have different units. This distinction is key. If you simply say you bake 20 cookies, that’s a quantity, but it doesn’t tell you anything about speed or efficiency. However, rates provide that important information by giving you a sense of how quickly something happens or how much of something occurs in a specific timeframe.

What are GRE Rate Problems?

GRE Rate Problems are a type of GRE quantitative reasoning question that tests your ability to analyse situations involving speed, distance, and time. Unlike classic work rate problems that focus on the amount of work completed, these problems deal with objects or people moving at certain rates. The goal is to find the total distance travelled, the time taken to cover a specific distance or the relative speeds of the moving objects.

These problems often involve scenarios like trains travelling towards each other, pipes filling a tank, or walkers covering a track. The key is to understand the relationship between rate, time, and distance. The formula, Distance = Rate x Time, becomes the foundation for solving these problems. By setting up the information about rates and distances travelled (or time taken) for each object, you can solve for the missing variable.

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GRE Rate-Time-Work Formula

The core formula for GRE work rate problems can be expressed in a few different ways, but they all boil down to the same principle: Rate multiplied by Time equals Work (R x T = W).

This formula essentially states that the total amount of work completed (W) depends on two factors: the rate (R) at which the work is done and the total time (T) spent working.

For instance, imagine a painter who can paint a room in 5 hours (rate). If they work for 3 hours (time), they will have completed a portion (work) of the room. The formula (R x T = W) allows you to calculate the completed portion (work) based on the rate (painting speed) and time spent painting.

The formula works both ways. If you know the total work (a fully painted room) and the time spent working (3 hours), you can solve for the painter’s rate (painting speed per hour). This flexibility is what makes the formula so important in the various work rate scenarios tested on the GRE.

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Average Rates on the GRE

On the GRE exam, you might come across specific rate questions called average rate problems. To solve these, you’ll need to know both the total distance travelled by an object and the total time it took to travel that distance. Armed with this information, we can determine the average rate using the following formula:

Average rate = (total distance) / (total time).

GRE average rate questions typically provide two distances and two times. These values can be directly plugged into the formula, like this:

Average rate = (distance one + distance two) / (time one + time two)

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Combined Worker Problems

Combined worker problems, a common type on the GRE, involve two or more entities working together to complete a task. We can solve these problems using a straightforward GRE math formula: the sum of the individual work rates (the amount completed per unit time by each entity) equals the total work rate needed to finish the job.

The total work itself can represent either a single unit (like painting a room) or a larger value, depending on the problem. While these entities collaborate, their work times may vary.

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6 GRE Rate-Time-Distance Questions with Solutions

Question 1: A train travels a certain distance at a speed of 60 mph. On the return trip, due to strong winds, the speed is reduced to 40 mph. If the total time taken for the round trip is 8 hours, find the distance travelled. 

Answer: Let D be the distance travelled (one way).

We know: Time = Distance / Rate

Time for the outward journey = D / 60 hours
Time for the return journey = D / 40 hours

Total time (given) = 8 hours

Therefore, D / 60 + D / 40 = 8

Simplifying the equation: (5D + 6D) / 240 = 8
Combining like terms: 11D / 240 = 8
Multiplying both sides by 240: 11D = 1920
Solving for D: D = 175 miles

The distance travelled is 175 miles.

Question 2: A car travels for 2 hours at a speed of 70 mph. It then encounters traffic and reduces its speed to 40 mph. If the total distance travelled is 180 miles, how long did the car travel at the slower speed?

Answer: Let T be the time spent travelling at 40 mph.

We know: Distance = Rate × Time

Distance travelled at 70 mph = 70 mph × 2 hours = 140 miles
Remaining distance (at 40 mph) = 180 miles - 140 miles = 40 miles

Distance travelled at 40 mph = 40 mph * T

Since the total distance is 180 miles, we can equate both expressions for distance: 40 mph * T = 40 miles

Solving for T: T = 40 miles / 40 mph = 1 hour

The car travelled for 1 hour at a slower speed.

Question 3: A boat travels upstream for 5 hours and then returns downstream in 3 hours. If the speed of the current is 2 miles per hour, what is the speed of the boat in still water?

Answer: Let the speed of the boat in still water be “B” miles per hour.

When travelling upstream, the boat’s effective speed is (B - 2) miles per hour, and when travelling downstream, it’s (B + 2) miles per hour.

Using the formula: Distance = Speed × Time

Upstream distance = (B - 2) × 5 = 5B - 10 miles
Downstream distance = (B + 2) × 3 = 3B + 6 miles

Since the distances are the same (going upstream and coming downstream), we can set them equal to each other:

5B - 10 = 3B + 6

Solving for B:
2B = 16
B = 8 miles per hour

So, the speed of the boat in still water is 8 miles per hour.

Question 4: A car and a bus start from the same point and travel in the same direction. The car travels at a speed of 60 miles per hour, while the bus travels at a speed of 45 miles per hour. If the bus departs 3 hours later than the car, how long does it take for the bus to catch up to the car?

Answer: First, calculate the distance the car travels in the 3-hour head start:

Distance = Speed × Time = 60 mph × 3 hours = 180 miles

Now, the bus needs to cover this 180-mile distance to catch up to the car. The relative speed between the bus and the car is:

Relative speed = 60 mph (car’s speed) - 45 mph (bus’s speed) = 15 mph

Now, use the formula: Time = Distance / Speed

Time = 180 miles / 15 mph = 12 hours

So, it takes the bus 12 hours to catch up to the car.

Question 5: Two cyclists start from the same point and travel in opposite directions. One cyclist travels at a speed of 12 miles per hour, while the other travels at a speed of 15 miles per hour. How far apart are they after 2 hours?

Answer: Since the cyclists are travelling in opposite directions, you can add their speeds to find the relative speed at which they are moving apart:

Relative speed = 12 mph + 15 mph = 27 mph

Now, use the formula: Distance = Speed × Time

Distance = 27 mph × 2 hours = 54 miles

So, the two cyclists are 54 miles apart after 2 hours.

Question 6: Lisa can paint a room by herself in 6 hours. Mike can paint the same room in 8 hours. Working together, how long will it take them to paint the room?

Answer: We can solve this problem by using the concept of rates. Lisa paints 1/6 of the room per hour (since she can complete the entire room in 6 hours). Mike paints 1/8 of the room per hour.

When they work together, their combined rate is the sum of their individual rates.

Combined Rate = Rate of Lisa + Rate of Mike
Combined Rate = 1/6 + 1/8

To find a common denominator, multiply the first term by 4/4 and the second term by 3/3:

Combined Rate = (1/6  4/4) + (1/8 × 3/3)
Combined Rate = 4/24 + 3/24
Combined Rate = 7/24

This combined rate represents the fraction of the room they can paint together in one hour. To find the total time they take to paint the entire room (which is 1 unit of work), we take the reciprocal of the combined rate.

Total Time = 1 / Combined Rate

Total Time = 1 / (7/24)

To simplify, multiply by 24/24:

Total Time = (1 × 24/24) / (7/24)
Total Time = 24 / 7

It will take Lisa and Mike 24/7 hours (or approximately 3 hours and 23 minutes) to paint the room together.

From the Desk of Yocket

GRE work rate problems require you to break down word problems into manageable parts. This involves identifying the workers, their rates (the amount of work completed per unit of time), and the total work required. Translating these elements into a Rate-Time-Work table allows you to see the relationships clearly. Once set up, you can use the formula Rate x Time = Work to solve for missing variables. This process strengthens your ability to analyze information, translate it mathematically, and solve problems efficiently. Practicing with GRE practice tests can further solidify your understanding and improve your performance on these types of questions.

While memorising a formula is helpful, the true strength comes from understanding the underlying relationship between rate, time, and work. Mastering these problems strengthens your analytical thinking and problem-solving abilities, which are applicable not just in the GRE but in various academic and professional settings. By applying these skills, you can approach complex situations with a structured and logical approach. Yocket Prep+ can provide additional practice and strategies to enhance your mastery of these concepts.