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Last time, I asked whether you know how—and when—to Work Backwards on Problem Solving problems. If you haven’t already worked through part 1 of this series, go do that and then come back here. I’ll wait. ☺
Here’s the second GMATPrep® problem from that set:
“If x² = 2y³ and 2y = 4, what is the value of x² + y ?
At first glance, this one does look like it might be a Working Backwards Problem Solving problem: the answers are pretty nice integers.
But! What’s the issue? Why wouldn’t we actually want to work backwards on this one?
Look at the question stem:
The strategy works really well when the question asks you for one discrete variable—not when it asks you for a combo (combination of variables), as in this problem.
Why? Try it out. Let’s say that x² + y = 6 (answer D). What are the values for x and y individually? Or even x² and y individually?
There are a lot of possibilities. I don’t want to try a lot of possibilities. I’m lazy; I want to get this Problem Solving problem done as easily as possible! So working backwards isn’t a good approach on this one.
What should I do instead? One of the equations is super easy: 2y = 4. Now I know that y = 2, and the problem is a whole lot easier to approach.
Plug y = 2 into the other equation and…wait.
Reflect a little more. What else do you need to know in order to answer the question?
x². Not x! So don’t solve for x; instead, solve for x². Don’t do more work than you have to do.
x² = 2y³
x² = 2(2)³
x² = 2(8)
x² = 16
Finally, find the value of x² + y.
x² + y
16 + 2 = 18
The correct answer is (E).
What did you learn on this Problem Solving problem? Think about your own takeaways before reading mine below.
One more question: Where do you think the wrong answers came from? They’re not random—they were chosen because they represent some kind of specific mistake that people are likely to make. If you can spot how the wrong answer was created, you are arming yourself against making those kinds of mistakes yourself.
In this case, let’s say that you did solve all the way for x. And then maybe you forget that they wanted x² + y—so you solve just for x + y. That’s 6, or trap answer (D).
And if you do solve for x, then you’d actually get x = 4 and –4. And –4 + 2 = –2, aka trap answer (B).
Trap answer (A) appears to come from adding –16 and 2, so that would be a couple of mistakes (because when you square –4, you should get a positive value, not a negative one).
I haven’t figured out where trap answer (C), 3, comes from. If you’ve got any ideas, share them in the comments.
The big takeaway for me on these wrong answers: Multiple of them revolve around solving all the way for x rather than solving for x². In other words, besides saving time, I’m also saving myself from making careless mistakes when I pause and make sure that I’m solving only for exactly what I need on this GMAT problem, not for what a math teacher would make me do.
Key Takeaways for Knowing the Code on Problem Solving Problems
(1) If the question asks for a combo (combination of variables), then working backwards probably won’t be a good strategy. Look for another angle.
(2) When a question does ask for a combo, think about how to solve directly for the entire combo or, at the least, for “chunks” of the combo. Don’t just auto-solve for all of the individual variables, the way you were taught in school.
(3) Turn that knowledge into Know the Code flash cards:
* GMATPrep® questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC. 📝
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Stacey Koprince is a Manhattan Prep instructor based in Montreal, Canada and Los Angeles, California. Stacey has been teaching the GMAT, GRE, and LSAT for more than 15 years and is one of the most well-known instructors in the industry. Stacey loves to teach and is absolutely fascinated by standardized tests. Check out Stacey’s upcoming GMAT courses here.