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Nine identical machines, each working at the same constant rate, can stitch 27 jerseys in 4 minutes. How many minutes would it take 4 such machines to stitch 60 jerseys?
First, take a deep breath. In this article, you’ll learn a methodical approach that will work on GRE rates problems every single time. On test day, it’ll be tempting to throw away your new habits and go back to old ones. Try to do the opposite. You’ve done all of this studying for a reason!
On problems like this, don’t try anything fancy. A lot of GRE test-takers will try to logically reason their way through this problem, saying something like “well, if 9 machines stitch 27 jerseys in 4 minutes, then 3 machines stitch 9 jerseys in 12 minutes…” That approach is valid but dangerous. Whenever you choose not to write something down, you’re taking away your ability to check your work for mistakes. (By the way, where’s the mistake in the logic described above?)
To start the problem, make a table. Your scratch paper should look like this:
Using the table, figure out how quickly a single machine is working. Solve the equation 9 * ? * 4 = 27 to learn that ?, the unknown value, equals 3/4. Add it to the table.
Once the first line is completely filled in, add a second line with the remaining information:
Finally, solve for the unknown value. In this case, we’re looking for the time. Solve the equation 4 * 3/4 * ? = 60 to find that ? equals 20. The answer is 20 minutes.
If you’re creating a cheat sheet for GRE Rates & Work problems, add the steps that we took to solve this one:
- Create a table
- Fill in the first line
- Find the rate of one worker
- Fill in the second line
- Solve for the unknown value
Always follow these steps, and you won’t go wrong. The advantage of filling out a table is that you can see which values you’ve calculated already and which values you still need to find. An unknown variable is just a blank space in the table.
There’s another, very similar type of problem in which the workers aren’t all identical. These problems look like this:
Jenny takes 3 hours to sand a picnic table; Laila can do the same job in 1/2 hour. Working together at their respective constant rates, Jenny and Laila can sand a picnic table in how many hours?
This is a ‘working together’ GRE rates problem, and the solution process is similar. Again, always start by creating a table. Since you aren’t worrying about identical workers, there’s no need to consider the rate of a single worker. Label the rows with the workers’ names, and fill in everything you know.
Once again, calculate the rate for each worker by solving the equations. In this case, Jenny’s rate is 1/3 tables per hour, and Laila’s rate is 2 tables per hour. (It feels a little silly to think in terms of ‘tables per hour’ or ‘violins per minute’, but it’s necessary in order to solve this type of problem). Then, create a third row to represent both workers’ combined efforts.
The rate of both workers combined is always the sum of their individual rates. That lets you fill in one more square in the table: Jenny and Laila’s rate when working together is 1/3 + 2, or 7/3 tables per hour.
Now there’s only one unknown. Solve the equation 7/3 * time = 1 to find that Jenny and Laila sand the table together in 3/7 hours.
Here are the steps:
- Create a table
- Fill in the first 2 (or 3, or 4) lines
- Find the rate of each worker
- Add the workers’ rates together to find the combined rate
- Fill in the last line
- Solve for the unknown value
Some GRE rates problems require creativity. Others require a methodical approach. These two Rates & Work problems fall into the second category. If you struggle with GRE Rates & Work, practice and review these strategies using simpler problems, then move on to tougher problems that might require other skills as well, such as unit conversions or percent calculations. The Rates & Work chapter of the 5lb. Book of GRE Practice Problems (where both of these problems came from) is a great place to start! ?
Chelsey Cooley is a Manhattan Prep instructor based in Seattle, Washington. Chelsey always followed her heart when it came to her education. Luckily, her heart led her straight to the perfect background for GMAT and GRE teaching: she has undergraduate degrees in mathematics and history, a master’s degree in linguistics, a 790 on the GMAT, and a perfect 170/170 on the GRE. Check out Chelsey’s upcoming GRE prep offerings here.