Solving GRE Problems in Multiple Ways to Build Flexibility
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Recently, my colleague Tom and I decided that, since we were teaching in adjacent classrooms, it might be fun to combine our classes and co-teach a lesson. Tom and I have very different strengths, both as test-takers and teachers. I love algebra, and I’ll always seek out an algebraic solution to a problem (even when this might not be the most efficient method—my strength is also a weakness). Tom prefers non-algebraic methods, like drawing diagrams or picking numbers. And our strengths inform what we emphasize in class.
So, for our joint lesson, we chose a number of GRE problems that could be solved in more than one way, and then took turns demonstrating each method. First, we each used the method we preferred (algebra for me, picking numbers for Tom), and then we switched and demonstrated the method we were less comfortable with. Here’s one of the GRE problems we used:
This kind of fraction translation problem can be really tricky for test takers, especially under time pressure. I’d probably do it using algebra. First, I’d assign variables:
John’s front lawn = F
John’s back lawn = B
Next, I’d start translating each sentence into algebra.
“John’s front lawn is 1/3 the size of his back lawn.” So, F = 1/3 B
“John mows ½ of his front lawn.” I’m looking for unmowed, though, which would also be ½ of the front lawn. Unmowed: ½ (1/3B) = 1/6 B
“John mows 2/3 of his back lawn.” So the part left unmowed is 1/3 of the back lawn, or 1/3 B.
Now I can add these together to get the total unmowed: 1/3B + 1/6B = 3/6 B, or 1/2B.
Am I done? Well, I’m looking for the fraction of the lawn left unmowed. That means I need part/whole, with the part being what’s unmowed (1/2B) and the total being the area of front AND back lawn. My total, then, is B + 1/3 B, or 4/3B. Now I have (1/2B)/(4/3)B. “B” cancels out, leaving me with a nice fraction: 3/8. And that’s the right answer!
Did that seem complicated or confusing? If so, don’t worry, because this isn’t the only way to solve this problem! Here’s how Tom would do it.
He’d notice that there aren’t any fixed quantities in the problem, only fractions. This means that, instead of assigning variables, he could also pick a number for the area of the back lawn and work with that number instead of doing algebra. Nice!
He’d also notice that I’m dividing the lawn into both halves and thirds. So, when picking an area for the lawn, it would help to have something that’s divisible by both two and three. Let’s say, then:
Back lawn = 6
This means that the front lawn = 2.
Now, half of the front lawn is left unmowed. So 1 unit is unmowed.
And 1/3 of the back lawn is left unmowed. Two units are unmowed.
My total area unmowed, then, is 3. And my total lawn area is 6 + 2 = 8. This gives me my fraction: 3/8.
Both methods work, and both allow us to arrive at the right answer. Which one you’d use on the test would depend on:
- Your own knowledge of yourself as a test-taker
- The nature of the problem
Some GRE problems are better done with one method or the other, but many can be done equally well with both. For this reason, it’s a good idea to practice both methods, so that you know what your options are and can make smart choices about when to use each method.
After Tom and I did our demonstration, we then split our students up into groups and had them work through several GRE problems together, practicing different methods of solving as they did. Since different people gravitate toward different methods, students were able to teach each other, depending on their own strengths and weaknesses.
Which brings me to my final point: having a study partner is a great idea. Not only do you help keep one another motivated, but you can also learn a lot from working with someone who thinks about things differently from you. Ultimately, you want to feel comfortable with both algebraic and non-algebraic solution methods and be able to deploy both efficiently and confidently when you take the test. ?
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Cat Powell
is a Manhattan Prep instructor based in New York, NY. She spent her undergraduate years at Harvard studying music and English and is now pursuing an MFA in fiction writing at Columbia University. Her affinity for standardized tests led her to a 169Q/170V score on the GRE. Check out Cat’s upcoming GRE courses here.