The GMAT Official Guide is a great teaching tool—all of the Verbal and Quant problems in the book are retired problems from real GMAT exams of yesteryear, so the “OG,” as we like to call it, is one of the best sources of practice problems for students.
Unfortunately, in many cases the OG leaves something to be desired in the explanations to those practice problems. The explanations are correct, and they’ll usually show most of the steps, but they can sometimes feel like they’re missing some key intermediate steps. And for students who are rusty doing arithmetic and algebra without a calculator, those intermediate steps can be the difference between getting Quant problems right in under 2 minutes and getting stuck or taking far longer than 2 minutes.
It reminds me of the “how to draw an owl” meme:
The OG explanations can sometimes feel like that, except they’re usually something like:
- Translate the problem into equations
- Solve the equations
For today, I’d like to focus on the second step: solving equations. For instance, how would you go about solving this equation without a calculator? I encourage you to grab a pen and paper and try to solve it right now (and time yourself!):
Did you get to an answer? How long did it take you?
An OG explanation might show that equation (after already having set up the equation) and then just go straight to the answer:
And it’s like, great—but how are you supposed to actually do that?
Unfortunately, when faced with an equation like this, many students go about solving for x in a far-from-ideal manner: they follow the same steps in the same order that they would take if they had a calculator, except they do the calculations longhand. So in this case, a student’s paper might look like this:
If your paper ever looks anything like that when doing GMAT Quant problems, then this column is for you.
Starting from the top left, the steps this student took to isolate x were:
- Multiply both sides by 35
- Calculate 1300*35 by stacking them and multiplying to get 45,500
- Divide both sides by 65
- Calculate 45,500/65 using long division (and some more multiplication in the right-hand margin to figure out that 65 goes into 455 7 times)
That’s correct, but it’s also incredibly inefficient. It takes a long time, it requires a lot of longhand multiplication and division, and it’s exhausting. Anytime you find yourself doing a lot of longhand multiplication and division on a GMAT problem, pause and ask yourself: is there a better way? 99% of the time, there is.
In this case, there are several key opportunities to simplify the problem that the student missed. The first is to think of this equation:
Like this instead:
Isolate the unknown variable x and think of the left side of the equation as x times the fraction 65/35 rather than 65x divided by 35. Now it’s easier to see that you can reduce the fraction: 65 and 35 are both divisible by 5, so with a little bit of mental math that fraction can be reduced to 13/7:
At this point it might still be tempting to multiply both sides of the equation by the denominator 7 and then divide by the numerator 13—that’s what you would do if you had a calculator. But without a calculator, it’s much more efficient to combine those two steps into one step by multiplying both sides by the reciprocal 7/13:
You can also think of that step as dividing both sides of the equation by 13/7. Either way, the 13/7 on the left side cancels out to 1, and we’re left with 1300 * (7/13) on the right side.
Now it becomes apparent that the 1300/13 are both divisible by 13, so 1300/13 can be reduced to 100:
And finally, all we have to do is multiply 100*7 and we’ve got our answer.
x = 700
So to recap, here’s what a more efficient test taker’s paper would look like to solve this equation:
Much easier, wouldn’t you say?
You might be thinking to yourself, “Okay, fine, but didn’t we get kinda lucky here? The 13 just happened to cancel out the 1300. If that hadn’t happened, then this wouldn’t be much faster than the original method.”
Fair point, but here’s the interesting thing: many GMAT Quant problems are specifically designed with these shortcuts, but the OG explanations almost never tell you when the writers are doing it. The writers want to reward test takers who can spot opportunities to solve a problem more efficiently, so very often an intimidating equation that looks like it will require a lot of calculations will turn out to be a mathematical house of cards, where almost everything cancels out with something else. But you have to look for those opportunities yourself. So we didn’t get lucky—we got rewarded for spotting the shortcut.
- If you find yourself doing a lot of longhand multiplication and division, pause and ask yourself: is there a better way?
- Reduce fractions whenever possible.
- Avoid using intermediate calculations to break up fractions; instead, multiply both sides by the reciprocal fraction and again look to reduce. 📝
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Ryan McGorman is a Manhattan Prep GMAT instructor based in New York, NY. He scored a 770 on the GMAT and has taught everything from SAT to GRE to public speaking and ESL. He earned his MBA at UCLA Anderson. Check out Ryan’s upcoming GMAT prep offerings here!