6 Mental Math Tricks For All Intents And Purposes

mentalmathtricksHaving a few mental math tricks up your sleeves will serve you well in a number of practical situations: drunk gambling, counting piles of money, the Mental Math Championships.  The list goes on for miles.  But practical purposes aside, mental math is fun, satisfying and it makes you feel and appear smarter than you probably are.  All wonderful things.  What follows is a list of mental math deceits, beginning on the novice end of the scale and culminating with tricks perhaps better labeled as high-flying acrobatics.  They might take some tinkering to get them running smoothly, but they will be by far the most impressive to innocent bystanders.  As the comeback kid of the decade, Ms. Britney Spears, says… You better work b*#!$.

Well, all I can offer you is a strong start.  But if you want to be a legend, you’re going to need something no one else has: the secrets of mental math.

Multiplying Large Numbers by Five

Ok, 10*5.  20*5.  50*5.  No big deal.  But what about 3672*5?  That’s what I thought.  Anyone can multiply small numbers by 5, especially is they’re multiples of 10.  But when you get to the big leagues, you’re going to need more than a vague image of your third-grade times tables in your head.  Check it out:

Let’s take that number I threw out there, 3672, and divide it by two.  If the result is a whole number, as it will be this time, then simply add a zero on the end, i.e. multiply by ten:

3672/2 = 1,836

1,836*10 = 18,360

Easy enough.  If the number has a remainder, you again multiply by 10.  Just make sure you don’t jump the gun with multi-digit remainders:

2,345/2 = 1,172.5

1,172.5*10 = 11,725

 

5,678.66/2 = 2,839.33

2,839.33*10 = 28,393.3

 

Calculating Tips

An ex-waiter once told me: if you can’t afford the tip, you can’t afford the meal.  And after reading this, you won’t be able to blame a cheap tip on poor math skills.  If you had good service and wanted to leave a 15% tip, the first thing to do is hash out 10%:

$68.00*.10 = $6.80

Now divide by two (this yields 5%):

$6.80/2 = $3.40

Add the two values together:

$6.80 + $3.40 = $10.20

But everyone knows 15% isn’t what it used to be.  For good service, more and more people are leaving 18-20% tips.  For 18%, take your 15% value, and add half of what you calculated for 5%:

$10.20 + ($3.40/2) = $11.90

That’s technically 17.5%, so you can throw a little bit on top (18% works out to $12.24).  But I don’t think the waiter will begrudge you thirty four cents.

Finding the Sum of the First n Integers

This is one question that frequently shows up on SATs and other standardized testing (you’ll need more than short cuts for the SAT; get detailed explanations for all levels of SAT math).  It’s a simple question if you know the trick, a time-waster if you don’t.

By “the first n integers,” I mean: if we take the number 16, we want to know the sum of 1 + 2 + 3 + 4 . . . + 16.  Fortunately, an easy equation exists to help us find the right answer:

sum = n(n+1) / 2

sum = 16(16+1) / 2

sum = 16(17)/2

sum = 8*17

sum = 136

In case you didn’t catch it, I divided the two from 16 to get 8, which makes the multiplication much easier.  Try a few yourself.  It’s even easier than it looks.

Square Any Number

This one is especially awesome, because squares are relatively common in every day life and proficiency with them is a sure sign of genius.  Let’s take a huge number: 9,999.  First, we need to figure out how much to add to get to the nearest ten.  In this case, 1.  So we add that to 9,999 to get 10,000, and we subtract that same number from 9,999 and get 9,998.  We multiply the two numbers, then add the square of “1”, or the difference to the nearest ten.  The equation looks like this, for a number, n, and difference to the nearest ten, d:

(n+d) (n-d) + (d^2)

(10,000) (9,998) + (1)

99,980,001

As long as you can move the decimal place accordingly, you can make short work of enormous squares.  Another quick trick for more reasonably sized squares adheres to the following equation for a number, n:

n^2 = (n*2) + ((n-1)^2) -1

That looks complicated, but all we’re doing is multiplying our number n by two, then adding it to the square of the next lowest number.  Then we just subtract by one.  So if you know 11^2 but not 12^2:

12^2 = (24) + (121) – (1)

12^2 = 144

We’re starting to get into things you’d see on case interview questions, where speed and efficiency are king.  Get lighting fast for your next interview. 

Turn Repeating Decimals into Fractions

Repeating decimals are like optical illusions.  You get sucked in and thinking is replaced with staring.  We’re going to turn the tables and replace all that staring with cheating.

We need a repeating decimal: 0.72727272…  Isolate the repeating number (72).  Notice this number has two places.  For every place, divide the repeating number by that number of nines (two places = two nines = 99):

0.727272… = 72/99 = 8/11

The sky is the limit for this trick.  Let’s look at something much larger: 0.189189189…

0.189189189… = 189/999 = 21/111

Multiplying by 11

You probably already know that to multiply numbers by 11, you multiply the number by 10, then add the original number.  So for 7, we get 70+7 = 77.  No surprises there.  But for larger, more awe-inspiring numbers, we need to streamline our process or we’re going to be left with an unmanageable addition at the end.

Let’s take the number 2,141.  Essentially, we are going to create a list of small numbers.  First, isolate the first and last numbers: 2 and 1.  You will want to leave them in their respective positions once we write the rest of the sequence.  To generate the other numbers, pair them in order.  A demonstration will show how easy this really is:

2,141*11 = (2) (2+1) (1+4) (4+1) (1)

But we aren’t multiplying them, as the dictation might suggest.  We are literally just writing the numbers as they appear, which gives us: 23,551.  Honestly, that’s probably the coolest one yet.  Before we wrap it up, you need to be prepared if your numbers add up to more than 10.

Out new number will be 3,672.  If we write it out like we did the first example, we get: (3) (9) (13) (9) (2).  But the answer is 40,392.  So how do we get there?  When the sum of two digits is greater than 10, carry the tens digit to the pair on the left.  So in our scenario, we carry the “1” from “13” left.  This turns “9” into “10”, so we leave the “0” and add the “1” to “3”.  This simple fix gets us to the right answer: 40,392.

Again, the sky is the limit.  I know math doesn’t always make sense, but these tricks are fairly straight forward and will always give you accurate results.  Something else that doesn’t make sense?  Easy advanced math.  But Wolfgang Riebe figured it out, and you can learn more techniques to unlock your genius from this mind-shift-master.