I hear this from a lot of you. Unfortunately, as I explained recently, having more time on the GRE wouldn’t actually help you get a higher score, since the GRE is a scaled test. So let’s leave the complaining to our competition, shall we? Instead of moaning about the clock, strive be as awesome as you can at solving problems. If you’re great, you’ll also be fast. Here’s a quantitative comparison that’s pretty simple, but also a nice illustratration of the fact that speed isn’t something that comes independently of problem solving skill.
This problem, like several I’ve been looking at recently, comes from our GRE Bootcamp event:
Quantity A: The sum of all integers from 9 to 29, inclusive
Quantity B: The sum of all integers from 12 to 30, inclusive
At a glance, the “math” way to solve this problem is time-consuming but direct: add up the sums in both columns, then compare. Since there’s an on-screen calculator on the GRE, some of your competition will solve the problem this way. And boy does it take a long time.
Let me be very clear: directly totaling both columns isn’t just a slow way to solve the problem. It’s a BAD way. Someone who solves the problem in this head-on, brute force fashion, then says to themselves, “I’m fine with the problems, it’s the timing that kills me,” is being dishonest with themselves. They are not ”fine with the problems.” They are very much unfine!
Instead, when you have to compare two quantities, start by eliminating what they have in common. If a quantity appears in both columns, then it isn’t helping either one to be bigger than the other.
Here, both columns include the range of numbers 12-29. Thus, totaling that range would be a waste of time. Ignore it and look instead at what’s different:
Quantity A: The sum of all integers from 9 to 11, inclusive
Quantity B: The of all integers from … never mind, it’s just 30!
And since 9 + 10 + 11 clearly equals 30, you can click choice (C) — “The two quantities are equal” — in under 10 seconds and score the point. That’s the beauty of the GRE: if you’re awesome, speed comes for free. Practice will get you there!
When a GRE quantitative problem features multiple ratios, many of you suffer headaches. This is because the “math” way of solving the problem is brutal, and students who don’t use logic will dive head-first into a morass of ugly substitutions, mistakenly assuming that the GRE is a math test. Here’s the kind of problem I’m talking about:
In a particular mixed candy bag, the ratio of Skittles to M&M’s is 4 to 5, while the ratio of Reese’s Pieces to M&M’s is 9 to 7. What is the ratio of Skittles to Reese’s Pieces?
The “math” way to do this problem is to set up two equations, solve one for M&M’s, and plug that value into the other one. If that sounds painful, that’s because it is. Don’t do this. Make a simple table instead:
S | M | R
4 : 5
7 : 9
Take a moment to confirm that you understand where the numbers above are coming from. They’re just a translation of the information in the word problem.
The question asks for the ratio of S to R. Can you just say it’s 4 to 9? No way. The value connecting them — the M — is different. It’s 5 in one ratio and 7 in the other. So, rewrite the ratios to make the M term the same in both, creating a kind of “bridge.”
Multiply the first ratio by 7: 7×(4:5) = 28:35
Multiply the second ratio by 5: 5×(7:9) = 35:45
Next, check out your new table:
S | M | R
28 : 35
35 : 45
Now you can just “walk across the bridge,” as it were — the ratio of S to R is simply 28:45. Try this technique on your next multiple-ratios problem and let us know how it goes!
Not too long ago, I used something called the “balance approach” to show you how to solve a mixtures problem. But the balance technique isn’t exclusive to mixtures. In fact, the most likely time for it to come up is when a problem deals with plain ol’ averages.
Here’s an example of the kind of GRE problem I’m talking about:
At a bowling tournament in which males and females competed, the average score of the participants was 154 points. If the average score of the 8 males was 148 points, how many females were in the tournament if the average female score was 158 points?
Most of your competition is going to try to use the average formula: average equals sum divided by number, or as I prefer to write it,
Average × Number = Sum
Using the above formula gives you this:
154(8 + x) = 8(148) + x(158)
There are a couple of problems here. One, you have to do some nasty arithmetic, such as 154×8 and 8×148. And perhaps more importantly, it takes a rather impressive feat of translation to set that baby up.
Try this instead.
The overall average is 154. Each man got 148 points, which is 6 points short of the average. There were 8 men, so altogether, they dragged down the average by 8×6 = 48 points.
This means the women need to make up a 48 point deficit. Each woman scores a 158, which is 4 points above the average. If each woman contributes 4 points to overcoming the 48 point shortfall, then there need to be 48/4 = 12 of them.
That sure was a lot easier than solving the equation. And if you’re a real critical thinking wizard, you might notice that even this was too much work. The ratio of the men’s deficit (-6) to the women’s surplus (+4) is 6 to 4, or 1.5. Thus, the ratio of women to men also needs to be 1.5, and that it is: 12÷8 = 1.5.
GRE teachers have a love/hate relationship with the calculator that testers get on their quantitative section. It can be useful in certain situations – if you have to multiply 372 by 754, for example, then by all means take advantage of the calculator to get a quick answer. By and large, though, we find that the majority of our students, even the ones who use a lot of math as part of their jobs or classes, overuse the calculator on problems that they could solve much faster by hand. It’s not because they’re unintelligent or lazy – given how ubiquitous calculators and spreadsheets are in our daily lives now, a lot of people just don’t remember how to do quick scratch-paper arithmetic. (You’ll notice that I didn’t say “mental” arithmetic – do not do any steps in your head! Write your work down. But that’s a topic for another day.)
So, to help remedy this problem, here are some quick arithmetic tricks that you can use to save yourself time and energy on the GRE:
- To divide by 4: Divide the number by 2, and then divide by 2 again.
- To multiply by 4: Flip the last trick. Double the number, and then double again.
- To multiply by 5: Multiply the number by 10, and divide by 2. (For integers, this just means adding a 0 to the end of the number and taking half of the result.)
- To divide by 5: Again, flip the previous tip. Divide the number by 10 (take away a 0 or move the decimal point one unit to the left), and double it.
- Once you’ve multiplying and dividing by 4 and 5 down, working with other numbers becomes simpler as well: Multiplying by 6, for example, just involves doubling a number and then multiplying that by 3.
- To quickly find percentages, multiply the number by the integer value of the percent, and use logic to determine where the decimal point goes. For example: If you need to find 12% of 300, first multiply 12 and 300: That gives us 3600. But we’re looking for a value that’s just a bit bigger than 30 (which is 10% of 300), so our answer must be just 36.
Applying these tricks may feel like a poor use of time at first, but if you practice by doing just a couple of calculations a day this way, by Test Day you’ll be a scratch-paper math whiz – able to outpace both the calculator and everyone else in the testing room!
BOB can eat a cheesecake in THREE minutes. His sister JENNY can eat a cheesecake in TEN minutes. How long will it take the two of them, eating TOGETHER, to eat three cheesecakes?
If you’re anything like I used to be, these problems drive you nuts. But I say “used to be” because after I started teaching for Kaplan, I learned that there is actually a dirt-simple plug-and-chug formula that’ll spit out the answer for you. It’s call the combined work formula, and it’s this:
A student in my class last night asked, “Why weren’t we taught this in high school?” Why, indeed. When I learned this formula was real, I wanted to go find my high school math teachers and smack ‘em.
So Bob and Jenny’s cheesecake addiction problem isn’t a hair-puller after all. Just plug their speeds into the formula…
…and remember that this is the GRE, so there’ll always be a trap. Here, remember that the question asks for the time to eat THREE cheesecakes, so multiply the above by 3:
And there’s your answer. In case you missed it in high school, don’t miss it now: the combined work formula is a mighty addition to your arsenal on Test Day!