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In my last blog post, we practiced using variables to solve Quant word problems—and we solved some problems without using variables, too. The big takeaway: you don’t have to start every word problem with a tidy little list of variables and equations! It’s okay to focus on the numbers in the problem first. However, variables are sometimes the key ingredient to getting a GRE problem right. In this article, we’ll try using variables to solve some tougher GRE Quant word problems.
Here’s one of my favorite problems from the 5lb. Book of GRE Practice Problems. Give it a try before you keep reading:
The yoga company Yoga for Life offers 45-minute classes at $12 per class. If the number of minutes Randolf spent doing yoga this month was 132 greater than the number of dollars he paid, how many classes did he attend?
Let’s treat this like a Numeric Entry problem, meaning that you don’t have any answer choices to help you out. (If you did, Backsolving would probably be the way to go!) There also isn’t an obvious way forward that only uses numbers. This is where a lot of people get stuck. You might start with the first sentence, and write down something like “45m = 12”, or “45 = 12c”. But are those actually the right equations? Unfortunately, no. In order to solve this one, you need to be extremely clear about what your variables represent.
Let’s say that m represents the number of minutes Randolf spent doing yoga this month, and c represents the number of classes he took. That’s actually the first thing I’d jot down on my scratch paper.
Now we can start creating equations. Or can we? Our first guess, 45m = 12, would be read like this:
“45 times the number of minutes equals 12.”
That doesn’t make sense—and by the way, reading your equations back to yourself in plain English is a good way to make sure they make logical sense. The number 12 doesn’t have anything to do with the number of minutes. It’s a number of dollars. Scanning the rest of the problem, the total number of dollars is going to be important. We’ll need a variable for that, as well.
Okay, back to that first sentence. You might be tempted to write something like “45m = 12d” at this point. Read it back to yourself first! There’s no reason to multiply the number of minutes by 45, or to multiply the number of dollars by 12. Here’s where you actually need to go next:
If you know how long one class is, you can relate the number of classes to the total amount of time.
The same is true for the cost of each class. If you know how much one class costs, you can relate the total number of classes to the total amount of money spent. In other words:
You can express both m and d in terms of c! The total number of minutes is 45 times the total number of classes. The number of dollars is 12 times the number of classes.
Finally, use this information to write a second equation and wrap up the problem.
Solving that equation tells you that Randolf took four yoga classes this month.
Okay, let’s focus in. Sometimes, in GRE word problems, the text tells you about a cost per unit, or time per unit, or something similar. In this problem, we learned about the cost per class and the time per class. A problem could also tell you about miles per hour, dollars per gallon, or even—as in the following problem—calories per doughnut.
Krunchy Kustard sells only two kinds of doughnuts: glazed and cream-filled. A glazed doughnut has 200 calories, and a cream-filled doughnut has 360 calories. If Felipe ate 5 doughnuts totaling 1,640 calories, how many were glazed?
This problem gives you two pieces of information: the total number of doughnuts and the total number of calories. It also tells you the number of calories per doughnut, which is what will let you relate the two pieces of info and create good equations. Try it out before looking at my scratch work:
Got it? Okay, let’s try one last problem that gives you information in a similar way. This one is trickier!
When traveling on the highway, a certain car averages 60 miles per hour and uses 2 gallons of gasoline per hour. When traveling in the city, the same car averages 30 miles per hour and uses 1.5 gallons of gasoline per hour. If the car uses a total of 8 gallons of gasoline over the course of a 210-mile trip, how many hours of the trip were spent traveling on the highway?
Try it on your own first. If you get stuck, look back at the two previous problems. This one is very similar. Your variables should be c, “city hours,” and h, “highway hours.”
Ready? Here’s how to approach it:
- Since the car spends c hours traveling in the city, it goes through 1.5c gallons of gas during that time, and travels 30c miles.
- Likewise, while the car is on the highway, it uses 2h gallons of gas and travels 60h miles.
- The total amount of gas used is 8 gallons, so 8 = 1.5c + 2h.
- The total distance traveled is 210 miles, so 210 = 30c + 60h.
Finish up by solving those two equations—you’ll find that the car spent 2.5 hours on the highway.
Look out for word problems like these ones in the future, and don’t let the GRE confuse you! If a problem tells you about the number of calories in one doughnut, you can convert that to the total number of calories. If a problem tells you how many miles a car travels per hour, you can convert that to find the total number of miles traveled. These problems might be intimidating, but you can solve them the same way every time—be consistent, and you’ll be well on your way to mastering GRE Quant. 📝
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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 170Q/170V on the GRE. Check out Chelsey’s upcoming GRE prep offerings here.