GRE Quantitative: Rates and Work Question Practice

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On the GRE, Rates and Work questions may appear in any of the Quantitative question formats: Multiple Choice, Numeric Entry, or Quantitative Comparisons. A “rate” is anything per anything (miles per hour, laps per minute, gallons of paint per square inch of wall, etc.).
In the meantime, here are two formulas you should memorize to get these types of questions correct on your GRE test:

  • The first GRE formula to memorize before your GRE test is: D = R x T. This stands for Distance = Rate x Time. It can also be rearranged as Time = Distance / Rate or as Rate = Distance / Time.
  • The second formula you’ll want to know is: Average Rate = Total Distance / Total Time. Average Rate may have the word “average” in it, but remember that this is an entirely different concept from mathematical mean. Let’s look at an example question:

Let’s review some practice questions:
 

GRE Quantitative: Rates and Work Practice Question


1. Joanne drove 80 miles to see her mother. It took her 4 hours to get there. Then, she left her mother’s and drove another 40 miles to visit her aunt, but this time went 40mph. What was her average speed for the whole trip?


Remember Average Speed = Total Distance / Total Time.
Joanne traveled 80 miles + 40 miles so the Total Distance was 120 miles. She drove for 4 hours + 1 hour (since 40 miles at 40mph would only be 1 hour) so the Total Time was 5 hours. 120/5 = 24.
Therefore, the average speed for the whole trip was 24 mph. Think of Average Speed as a weighted average. Joanne spent more time going 20mph than 40mph, so it makes sense that the Average Speed would be closer to 20mph.



Let’s try another practice question.

2. Marion spent a day on a sightseeing trip in Tuscany. First she boarded a bus which went 15mph through a 30 mile section of the countryside. The bus then stopped for lunch in Florence before continuing on a 3 hour tour of the city’s sights at speed of 10mph. Finally, the bus left the city and drove 40 miles straight back to the hotel. Marion arrived back at her hotel exactly 2 hours after leaving Florence. What was the bus’s average rate for the entire journey?


To find the “Average Rate” of the bus, we know we will need to find the Total Distance and the Total Time, so let’s see how we can use the D = R x T formula to find the missing info.
For the first part of the trip, we know that 30 miles = 15mph x T, so we know that T = 2 hours.
For the middle part of the trip, we know that D = 10mph x 3 hours, so we know that D = 30 miles.
For the last part of the trip, we know that 40 miles = R x 2 hours, so we know that R = 20mph.
Now we can find the Total Distance and the Total Time.
Total Distance = 30 miles + 30 miles + 40miles = 100 miles.
Total Time = 2 hours + 3 hours + 2 hours = 7 hours.
So the Average Rate = 100 miles/ 7 hours = 14.28mph.



Converting Rates on the GRE


Some GRE rate questions will be presented as Quantitative Comparisons and will require conversions. You won’t be required to know complicated conversions (such as liters to gallons) but you must know a few basics chronological ones. There are 60 seconds in 1 minute, 60 minutes in 1 hour, 24 hours in a day, and 365 days in one year. For measurement, it’s enough to know that a foot has 12 inches.
Let’s look at this question:

3.

Column A Column B
The number of seconds in 2 hours The number of days in 20 years

A. Quantity A is greater.
B. Quantity B is greater.
C. The two quantities are equal.
D. The relationship cannot be determined from the information given.


2 hours = 120 minutes = 7200 seconds
20 years  = 365 days x 20 = 7300.
Even without considering leap years, (B) will be greater.



Sometimes the GRE will present a work problem involving the amount of work that can be done individually, and then combined. Remember that the amount of a job that an individual can complete in one hours is always the reciprocal of the number of hours it takes to complete the full job.
For example, if Sheila takes 4 hours to clean her room, then she can clean ¼ of her room in 1 hour. If Sheila’s mom can clean her room in 3 hours, then Sheila’s mom can clean 1/3 of the room in 1 hour. Working together, they will clean ¼ + 1/3 = 7/12 of the room in 1 hour. As a result, it will take them less than 2 hours to finish cleaning when they work together. Remember to ADD the individual rates to find the COMBINED rate. Let’s try one more word problem to put our formulas to the test!

4. Tracey ran to the top of a steep hill at an average pace of 6 miles per hour. She took the exact same trail back down. To her relief, the descent was much faster; her average speed rose to 14 miles per hour. If the entire run took Tracey exactly one hour to complete and she did not make any stops, how many miles is the trail one way?


For the way up the hill, we know that D = 6mph x T.
For the way down the hill, we know that D = 14mph x T.
Since we went know that the distance up the hill was the same as the distance down the hill, we can pick a number for D. Let’s choose “84” since it is a multiple of both 6 and 14.  If 84 = 6mph x T, then we know that T = 14 hours. If 84 = 14mph x T, then we know that T  = 6 hours.
Now we can use another formula, the Average Rate formula, to find the average speed for the WHOLE trip. Average Rate = Total Distance / Total Time
Using our Picked Number of 84, we know that the Total Distance traveled would be 168 miles. The Total Time is 14 hours + 6 hours = 20 hours.  So the Average Rate = 168 miles / 20 hours = 8.4 mph.
It doesn’t matter that Tracey didn’t “really” go 168 miles, or that we know she didn’t “really” go 20 hours. We Picked a Number just so that we could find the ratio of the Total Distance to the Total Time in order to calculate the Average Rate of the ENTIRE journey.
Now that we have found the Average Rate for the whole trip, we can plug it in to the “DIRT” formula to find the ACTUAL distance for the entire journey.
D = R x T
D = 8.4mph x 1 hour
We know that T = 1 hour because the problem told us so. Therefore, the actual distance for the entire trip was 8.4 miles. The problem asks how many miles the trail was one way. 8.4 / 2 = 4.2. The answer to the question is 4.2 miles.
You could also solve this problem in other ways, including using a system of equations and substitution, but it’s nice to know that you can pick a number for the Distance traveled and use it to find the Average Rate for the whole journey!