Using Smart Numbers for GRE Quant
Here’s a quick cheat sheet on how, when, and why to use Smart Numbers to solve GRE Quant problems.
What is Smart Numbers?
Smart Numbers is a strategy for certain GRE Quant problems, usually word problems. It’s not a guessing method—in other words, using Smart Numbers will give you the exact right answer, just like doing algebra will.
When can you use Smart Numbers on GRE Quant?
You’ll decide whether to use Smart Numbers by looking at the answer choices (so, it’s most often useful on Discrete Quant problems, which have answer choices!).
If you see the following in the answer choices, you can definitely use Smart Numbers:
- Expressions with variables in them, such as 3x or 4y + z.
You can also usually use Smart Numbers if you see the following in the answer choices:
- Percents
- Ratios
- Fractions
If you see percents, ratios, or fractions, here’s how to make the decision. Read the whole problem, and decide whether you’re dealing with specific numbers, or just with relationships between numbers.
For instance, does the problem say that x equals 12, or that Beryl has sixteen cats? Those are specific numbers, and you probably can’t use Smart Numbers.
On the other hand, if x is 50% more than y, or if Beryl has twice as many cats as Jane, those are relationships—and you probably can use Smart Numbers.
There are a few other special situations, so I’ll also give you a rule that covers everything—although it takes a little bit more thinking to apply it. If a GRE Discrete Quant problem doesn’t tell you the numbers, but just tells you how they relate to each other, you can use Smart Numbers. If it does tell you specific numbers, you can’t.
How does Smart Numbers work?
Suppose you’ve decided to use Smart Numbers because there are variable expressions in your answer choices. For instance, the problem looks like this:
If a, b, c, and d are consecutive integers and a < b < c < d, what is the average (arithmetic mean) of a, b, c, and d in terms of d?
A) d – 5/2
B) d – 2
C) d – 3/2
D) d + 3/2
E) (4d – 6)/7
In this situation, start by choosing numbers that fit all of the facts the problem gives you. In this one, the four numbers you choose have to be consecutive, with a being the smallest, and d being the largest.
As long as the numbers fit the facts, you should use the easiest numbers you can think of. For this problem, let’s go for 1, 2, 3, and 4.
The next step is probably the most important one: everywhere you see a variable in the problem—including the answer choices!—replace it with the number you chose. You can use a combination of mental math and scratch work to do this, depending on how complex the problem looks.
By the way, during this step, you should forget about the phrase “in terms of d.” “In terms of” only matters when you’re using variables. Since we’re replacing our variables with numbers, we can just drop it.
Here’s what that problem would look like, once we’re finished with this step:
If 1, 2, 3, and 4 are consecutive integers and 1<2<3<4, what is the average of 1, 2, 3, and 4?
A) 4 – 5/2
B) 4 – 2
C) 4 – 3/2
D) 4 + 3/2
E) (4*4 – 6)/7
Next, answer the question. What is the average of 1, 2, 3, and 4? It’s 2.5.
Which of the answer choices equals 2.5? Only (C) does. (By the way, you can often figure this out without doing too much math—for instance, you should eliminate (B) quickly, since it won’t result in a decimal.)
Let’s try another one. This time, suppose you’re using Smart Numbers because you noticed percents in the answer choices. Your problem might look like this:
Aloysius spends 50% of his income on rent, utilities, and insurance, and 20% on food. If he spends 30% of the remainder on video games and has no other expenditures, what percent of his income is left after all of the expenditures?
A) 30%
B) 21%
C) 20%
D) 9%
E) 0%
Pick a number that fits everything you’re told in the problem. This problem doesn’t really give us any constraints on the number—except that it’s a dollar amount, so it shouldn’t be negative—so we can pick more or less any number we want. Let’s say that Aloysius’s income is $100.
You don’t have to replace the variables with numbers in this scenario, because there aren’t any variables! If the problem only has percents or ratios, not variables, you can skip that step. Go right ahead and solve the problem.
50% of $100 is $50, and 20% of $100 is $20. That leaves $30 remaining. Aloysius spends 30% of that $30, or $9, on video games. His total expenditures are $50+$20+$9, or $79, with $21 left over. Since $21 is 21% of his original income, the right answer is (B).
Why should you use Smart Numbers?
In some situations, using Smart Numbers takes more time than just doing the algebra. If you’re fast and confident with algebra, there will be problems where you’ll save time by “just doing the math.” However, there are other advantages to using Smart Numbers:
- It’s easier to check your work with numbers than with variables.
- It makes it easier to convert between different units. It’s much easier to convert 100 pennies to dollars than to convert 4x pennies to dollars.
- It makes it easier to work with percentages. I know that 3 is 50% of 6, but it’s not nearly as obvious that 3xy is 50y% of 6x.
- It’s often an easier way to solve a very tough word problem. If you’re having a hard time setting up equations based on a word problem, it may become clearer when you try using specific numbers.
However, I do have one warning: don’t think of Smart Numbers as a last resort! If you wait until you’ve already spent two minutes on the GRE Quant problem, using Smart Numbers isn’t going to help you. Try using it first—after all, there’s no rule saying you have to try algebra before you can do something else. On the GRE, you’re free to use whichever approach works, even if your middle school algebra teacher would disapprove! ?
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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.