Mental Math Magic (Part 2)

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Manhattan Prep GRE Blog - Mental Math Magic (Part 2) by Neil Thornton

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In my last article, I gave you some time-saving basic arithmetic to memorize and a few tools to calculate more efficiently, using a combination of your brain and your scratch paper.

Today I’m going to throw few fun mental math “tricks” your way. Again, you could always pop out your calculator or do long division and multiplication on your scratch paper, but learning to multiply numbers in your head can be a massive time saver, as well as a good way to double-check what you do put into the calculator.

Multiplying by 5

Start with the simple fact that 5 is half of 10:

5 x 2 = 10, and therefore 5 = 10/2.

If you want to multiply by 5 easily in your head, try multiplying by 10 and dividing by 2. Or a bit easier: find half of it and then and multiply by 10.

8 x 5 = ?

Think: half of 8 is 4, and 4 times 10 is 40. So 8 x 5 = 40! You should have that memorized, but it’s a good example to start.

240 x 5 = ?

Half of 240 is 120, and 120 x 10 is 1200, so 240 x 5 = 1200.

Try it on your own, and if you must, use a calculator to check your work:

210 x 5 = ?
54 x 5 = ?
62 x 5 = ?

By the way, this method works with odd numbers, too!

19 x 5 = ?

Half of 19 is 9.5, and 9.5 x 10 = 95. So 19 x 5 = 95.

37 x 5 = ?

Half of 37 is 18.5, and 18.5 x 10 = 185, so 39 x 5 = 185.

Dividing by 5

By the same principle, dividing by 5 is the same as multiplying by 2/10. So to divide by 5, double the number and then divide by 10.

210 / 5 = ?

Double 210 is 420. Divide by 10 = 42. So 210 divided by 5 is 42!

300 / 5 = 60

This works with numbers that don’t end in 0, too!

76 / 5 = ?

Double 76 to get 152, divided by 10 is 15.2, so 76 / 5 = 15.2!

43 / 5  = ?

Double 43 to get 86, divided by 10 is 8.6, so 43 / 5 = 8.6.

Work these a few times, and be careful not to get the methods mixed up! Think before you start: Is it 10/2 or 2/10?

Multiplying by 9

Here’s a trick I learned in 3rd grade:

Hold your hands up in front of you, palms facing you. If you want to multiply any number (from 1 to 10) by 9, just hold down that finger. The number of fingers to the left of your held-down finger is the 10’s digit. The number to the right is the units digit.

9 x 3 = ?

Hold down your third finger. You’ll have 2 fingers to the left, 7 fingers to the right. So 9 x 3 = 27.

9 x 6 = ?

Hold down your 6th finger. You’ll have 5 to the left, 4 to the right, so 9 x 6 = 54.

Once you do this a few times, you’ll be able to visualize the approach and knock out your multiples of 9 with ease.

Multiplying by 11

In my last article, I taught you a “right-to-left” approach that will work for anything multiplied by 11.

11 is just 10 + 1. You can take the number, multiply it by 10, and then add the number, and you’re done.

240 x 11 = ?

Think: 240 x 10 = 2400, 240 x 1 = 240. Add them, and therefore 240 x 11 = 2640.

Even faster: If you’re multiplying a small two-digit number by 11, you can take a cute shortcut.

1. Imagine the 2-digit number, and then “stretch them out” so there’s a space between the two digits. (Change 24 to 2_4 in your head.)
2. Add the 2 digits. (2 + 4 = 6)
3. Put the sum in between the two digits (264)

35 x 11 = ?

Imagine the digits with a space between them: 3_5.

Add 3 + 5 = 8.

Put that 8 in between the two digits, and get 385.

35 x 11 = 385

27 x 11 = ?

Think: 2_7.

2 + 7 = 9
27 x 11 = 297

If the two digits add up to something more than 10, don’t panic. Just add that extra 1 to the hundreds digit and squeeze the other number in-between.

87 x 11 = ?

Think: 8_7.

Add 8 + 7 = 15.

Add 1 to the 8, squeeze the 5 in-between.

87 x 11 = 957

Cool, eh?

Any other tricks you have? How do you visualize complex arithmetic? Please comment below! 📝


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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.

  1. Buhari Ibrahim September 9, 2017 at 3:36 pm

    Hey Neil, thanks for the time-saving tricks, 1 & 2, for multiplication and division. I took the GRE on Aug. 28th, and got 149V, 155Q, and 3.5 AW. I am planning to retake it in the next two or three months.

    Anyway, I was reading your post (loved it) and I also sort of saw an interesting pattern in multiplying by 9 that I thought will be worth sharing.

    9×1 = 09
    9×2 = 18
    9×3 = 27….

    What I saw is that, when you multiply any number by 9, for the first ten, the tens digit of the result is that number (multiplier) less 1, and the units digit is whatever adds to the tens digit to give 9. So for example, 9×1 = 09; 0 is 1 less than 1, and 0+9 = 9, so units digit is 9. Similarly, 9×8 = 72; 7 is 1 less than 8, and 7+2 = 9, thus 2 is the units digits.

    You just have to know 9×11 = 99, and another pattern develops, from 12 through twenty. However, this time around, the sum must be 18 (9×2 since we’re in the teens), instead of 9. And, the hundreds and the tens digits together should be 2 (not 1) below the multiplier.

    So, for example, 9×12 = 108; the hundreds and tens digits together, 10, is 2 below 12 (multiplier), and 8 (the units digit) adds to 10 to make 18. Similarly, 9×17 = 153; 15 is 2 below 17, and 3 adds to 15 to make 18.

    I hope I am making some sense here. Let me know what yo think of this.

    P.S. Any tips on doing better on the quant will be much appreciated.

    Thanks

    Buhari