# Division Shortcuts

Hey, I’m back. I gave up blogging because it seemed so heavy. But maybe I just need to lighten up, and keep blogging.

So it’s math lite time. You know, things one might find interesting, and maybe even useful.

## What is This Division Shortcut?

Q: Quick, quick, quick: Tell me what’s 1500 divided by 25? No calculator allowed. No pencil or paper.

A: How many quarters are in $15? Well, let’s see: There are 4 quarters in each dollar, so in$15 there must be

$15\&space;dollars&space;\cdot&space;4\&space;quarters\&space;per\&space;dollar&space;=&space;60\&space;quarters.$

Q: What are you talking about? I have to figure out 1500 divided by 25.

A: OK. There are 1500 cents in \$15. So 1500 cents divided by 25 cents per quarter equals 1500 divided by 25. That’s 60.

Here’s how a math teacher might explain it:

You need a quick way to divide 1500 by 25. Notice that

$25=100/4.$

So any number n divided by 25 equals

$\frac{n}{25}=\frac{n}{100/4}=\frac{4n}{4(100/4)}=\frac{4n}{100}$

Then, to divide a number n by 25, multiply by 4 and divide by 100. That is more steps than simply dividing, but the steps are easier:

$1500/25&space;=&space;1500\cdot&space;4/100=15\cdot&space;4=60.$

## When Does It Work?

This doesn’t work for a whole lot of divisors. It’s true that you can turn any division problem into a multiplication problem — just multiply by the divisor’s reciprocal. But in most cases that doesn’t make things easier. For example, what is $1500/17?$ Well, that’s $1500&space;*&space;1/17.$ Doesn’t  help. That’s because there is not a whole number of 17’s in 100.

But there is a whole number of 25’s in 100: 4.

There is also a whole number of 20’s in 100: 5. For example how many 20’s are in 2500? Use the same method:

$2500/20&space;=&space;2500*(5/100)&space;=&space;25*5&space;=&space;125.$

It also works for division by 10 and by 50:

 Divisor (n) n‘s reciprocal Number of n in 100 1500/n 10 1/10 $100\cdot&space;1/10=10$$100\cdot&space;1/10=10$ $1500/10=1500\cdot&space;(1/10)=150$$1500/10=1500\cdot&space;(1/10)=150$ 20 1/20 $100\cdot&space;1/20=5$$100\cdot&space;1/20=5$ $1500/20=1500\cdot&space;(5/100)=75$$1500/20=1500\cdot&space;(5/100)=75$ 25 1/25 $100\cdot&space;1/25=4$$100\cdot&space;1/25=4$ $1500/25=1500\cdot&space;(4/100)=60$$1500/25=1500\cdot&space;(4/100)=60$ 50 1/50 $100\cdot&space;1/50=2$$100\cdot&space;1/50=2$ $1500/50=1500\cdot&space;(2/100)=30$$1500/50=1500\cdot&space;(2/100)=30$

In this table, 1500/equals the number of 10s, 20s, 25s, or 50s in 1500.

This division shortcut works often enough so that if you pick your moment carefully you can amaze your friends with your speed at calculating.

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