Mental Math Magic

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Manhattan Prep GRE Blog - Mental Math Magic by Neil Thornton

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Quick!
What is 12 x 9?
What is 9³?
What is the square root of 196?
What is 95 – 37?

How long did it take you to figure those out using mental math? Did you need a pencil or a calculator? Are you 100% sure you’re right? Do you know a way to check your answers? Do you still not know the results? (Okay, lazybones: 108, 729, 14, 58).

If you’re like most students, your major concern about the Quant section of the GRE is time. You could get way more questions right if you only had more time.

The GRE is never as easy as knowing 12 x 9 = 108. However, a single tough GRE problem may require a handful of little calculations, each of which can eat up precious time. If you know 12 x 9 = 108, you’re done in seconds. If you have to pop out the calculator and key in each number carefully, you’ve added 30 seconds for each calculation, or several MINUTES to each problem—time you just don’t have.

Phase 1: Memorize Your Basics

Yes, you have a calculator, but you can’t rely on it. You MUST know your basic times tables. 2 x 1, 2 x 2, 2 x 3…7 x 8, 7 x 9…all the way to 12 x 12 (at least).

Write the numbers 1 through 12 across the top of a piece of paper. Then 1 through 12 down the left-hand side. Get to work filling in the grid with all those multiples. Next week, mix up the numbers, or fill the spaces in randomly. Once you have them locked, add 13s and 14s to the mix. Can you count up by 17s? 19s?

Make flashcards of the tough ones. Download iPhone apps. Run through your 7’s as you’re walking to work. 42…49…56…63…

Other Basics to Know

Addition of all single-digit numbers: 1+1 to 9+9
Subtraction of all “teens”: 11 — 2, 11 — 3… 18 — 9, etc.

Rules of divisibility: Can you look at a number and tell whether it’s divisible by 2, 3, 4, 5, 6, 8, 9, or 10? Check out our Number Properties Strategy Guide for all of these rules.

Perfect squares: 1², 2², 3²… all the way to 20²
Perfect cubes: 1³ to 10³

Powers of 2: 2¹ to 210
Powers of 3: 31 to 36 (Note that 36 is the same as 93. Why would that be?)
Powers of 5: 5to 54

Lock these in and you’ll be saving 10 seconds here, 30 seconds there, all of which will add up to finishing way more problems than your competition. Even if you don’t have everything perfect, all the effort will be worth it. Other people will see 243 and not know what to do (is it prime? What?). You’ll see 243 and think “Hmm, that looks like one of those powers I memorized. Okay, the digits add up to 9, so that’s a multiple of 9…so it must be a power of 3. Which one? Well, 81 is 34, so 243 must be 35!”

Phase 2: Mental Math Magic

After you have the basics memorized, you can get to all the ways to impress your friends (if they’re not already impressed by 14 x 14 = 196). I’ll show you more tools and tricks for mental math magic in future articles, but let’s learn the first principle of quick mental math: Go Left to Right.

When you learned so-called “long” multiplication in grade school, you learned a RIGHT to LEFT technique that is practical on paper (and very useful on the GRE), but nearly impossible to perform in your head. Take 23 x 9:

23
x 9

Notice, old-school multiplication goes right to left. “What is 3 times 9? 27? Okay, carry the 2… What is 2 times 9? 18, add the 2 to get 20. So 207.” That’s cool. Good to know, but you can actually do this in your head (with maybe a little help from your pencil) if you go LEFT to RIGHT instead.

23 x 9.

Think of 23 as 20 plus 3. Now multiply things left to right. 20 x 9 is 180. Store that or write it down. Now 3 x 9 = 27. Add 180 to 27 and you get 207. Way easier. No need to “carry” anything.

Let’s try a tougher one—253 x 4.

Think of that 253 as 200+50+3.

200 x 4 = 800
50 x 4 = 200
3 x 4 = 12

Add ‘em up and you get 1012!

Phase 3: Check Your Work

Remember 12 x 9? If I don’t have it memorized, I can use the left to right technique:

10 x 9 = 90
2 x 9 = 18

Therefore 12 x 9 = 108. But how can I be relatively sure that I didn’t make a mistake?

Ballpark

First, I make sure I’m in the right range by rounding the numbers to easy things (10, 100, 25).

9 is a little less than 10. 12 is a little more than 10. 10 x 10 = 100, so I know I’m looking for a number close to 100. 108 is close enough.

23 x 9? 9 is close to 10, so I’m looking for an answer less than 230. 207 is close enough.

253 x 4? 253 is very close to 250, so I’m looking for something close to 1000. 1012 is close enough.

Check Divisibility

Multiples of 9 are a little magical. To check whether a number is divisible by 9, add all the digits together. If you get 9 or a multiple of 9, you know you have a multiple of 9.

12 x 9 = 108. Add those digits together: 1+0+8 = 9. I’m pretty sure I have it right. If I get 107, the digits add up to 8, so I know I made a mistake.

Wait, does 9³ = 719? Or 721? Or 729? The only one in which the digits add up to a multiple of 9 is 729. 7+2+9 = 18, so that’s probably it. Also, 10 x 10 x 10 is 1000, so 729 is in the right ballpark. 243 would be way too low.

Practice and Play

You can’t spend all day every day doing GRE problems without going a little crazy, but you can have fun throughout the day practicing your mental math. As you’re walking and driving, do your times tables, run through your perfect cubes. Add up groceries in your head. Estimate the tax. Practice ballparking to divide the dinner check by the number of people at the table; estimate the tip.

Whenever you see a number (an address, an area code), double it, triple it, multiply it by 7. Check to see what it’s divisible by. (I used to live in New Orleans, area code 504. 504 is divisible by 2, 3, 4, 6, and 9. And 504 x 7 is 3528, or is it?)

A week or two of this kind of practice can cut your calculation time (not to mention your stress levels) in half or more. Do it! Have fun.

More to come! 📝


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Neil Thornton Instructor HeadshotWhen not onstage telling jokes, Neil Thornton loves teaching you to beat the GRE and GMAT. Since 1991, he’s coached thousands of students through the GRE, GMAT, LSAT, MCAT, and SAT and trained instructors all over the United States. He scored 780 on the GMAT, a perfect 170Q/170V on the GRE, and a 99th-percentile score on the LSAT. Check out Neil’s upcoming GRE course offerings here or join him for a free online study session twice monthly in Mondays with Neil.

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