### Mental Math Tricks & Hacks

People of a certain age (ask your parents, they’ll confirm) will remember a time when, if students complained about “no calculators allowed” math tests, the teacher would say that “you won’t always have a calculator in your pocket, you need to be able to do without.”

Well, the math teachers of yore were completely wrong about me not always having a calculator in my pocket. But it turns out that being able to do math in my head is actually super useful. If I need to calculate something like sales tax or a tip, it’s usually faster and easier to do it in my head than it is to pull out my cell phone.

From the perspective of a student, it is helpful in lots of ways. Plenty of tests are no-calculators-allowed, in whole or in part. They’re written so that the arithmetic involved isn’t too difficult, but it’s still helpful to be able to quickly calculate 15 x 20 in your head rather than writing longhand. And even on tests where calculators are allowed, doing it mentally is, again, often easier and faster once you get the hang of it.

Plus it makes a good party trick. You can really impress people, maybe even get a date. (I said “maybe.” It could happen.)

## Powers of 10

Multiplying or dividing something by 10, 100, or 1000 is where a lot of this begins. If you can count, you can multiply or divide by powers of 10 in your head. Just count the zeros and move the decimal point that many places — to the right if you’re multiplying, to the left if you’re dividing.

So, let’s say you are multiplying 568 x 100:

1. Count the zeros in 100. 1! 2! AH AH AH! (Sorry, channeled a little bit of the Count from Sesame Street there.)
2. Move the decimal place (not generally written with a number like 568, but it’s there) two places to the right, giving you 56,800.

Now, let’s try dividing 568 by 1000.

1. Count the zeros in 1000 — there are 3. (Not doing the Count joke again, don’t worry.)
2. Move the decimal place 3 places to the left, giving you 0.568.

## Multiplying and dividing by 5

To multiply by 5, first you divide a number in half, then multiply by 10.

Like so, if we want to do 36 x 5:

1. 36 ÷ 2 = 18
2. 18 x 10 = 180

Dividing? The opposite. Double, then divide by 10.

So, if we want to do 36 ÷ 5:

1. 36 x 2 = 72
2. 72 ÷ 10 = 7.2

## Multiplying a 2-digit number by 11

Let’s say we want to do it with, say, 34. Follow these steps:

1. Add the digits of your number together, in this case, 3 + 4 = 7
2. Stick the number you get in the middle of your starting number, so in this case, it would be 374 … and you’re done! 34 x 11 = 374

It’s slightly tricker if you get a number in step 1 above that’s 10 or above, but still quite doable. Let’s say we want to multiply 11 by 75:

1. We once again add our digits together and get 7 + 5 = 12.
2. So what do we do now that we have two digits? We stick the 2 in the middle, as before, and add the 1 to the first digit (in this case, 7). 7 + 1 is 8, so our answer is 825.

And if we want to move beyond 2-digit numbers, we multiply our number by 10 and add the starting number. Let’s do 819 x 11:

1. 819 x 10 = 8190
2. 8190 + 819 = 9009

## Squaring a multiple of 5

This works for any 2-digit number that ends in 5. 15, 25, all the way up to 95. All you do is multiply the 10s digit by one number higher, and then stick 25 onto the end of it.

Let’s solve 652 as an example:

1. Multiply the 10s digit — 6, in this case — by one number higher — in this case, 7. So, 6 x 7 = 42.
2. Stick 25 onto the end of 42, so you get 4225.

## Multiplying a big number by 9

This one is easy as multiplying by 10 (or, in other words, sticking a zero onto the end) and then subtracting your starting number. Let’s do 268 x 9:

1. 268 x 10 = 2680
2. 2680 – 268 = 2412

I’ll leave things there, but there are a ton of these tricks out there. They do take practice to really get the hang of them, but they’re worth it.

Plus, it is very useful on tests — more so than you might think. Sure, you might think, you showed us ways to multiply by 5 and by 9, but what if you need to multiply something by, say 7? Or 8⅔? I don’t have a quick and easy way to help you multiple 117 x 8⅔, but I can tell you thanks to one of my tricks above that 117 x 9 = 1053. So, I’d guess our actual answer would be a bit less than that, so … something near 1000? That’s generally more than enough to answer a multiple-choice question. Using my calculator, I can tell you that the actual answer is 1014, so I’d say a guess of “about 1000” is pretty darned good.

And like I said at the top, I guarantee you’ll be a hit at parties.

Nerdy, nerdy parties.

#### Michael DePalatis

##### Tutor

Mike grew up camping, skiing, and reading in the mountains of Western Maryland. (Yes, people actually live out there. There are several of us! Several!) He was a scientist from a young age, relentless in his pursuit of knowledge, even when his curiosity led to him flooding the basement at the age of four. (He maintains to this day that his experimental design was conceptually sound.) M...