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

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

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:

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

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 $1500 * 1/17.$ 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:

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

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	$100\cdot 1/10=10$	$1500/10=1500\cdot (1/10)=150$
20	1/20	$100\cdot 1/20=5$	$1500/20=1500\cdot (5/100)=75$
25	1/25	$100\cdot 1/25=4$	$1500/25=1500\cdot (4/100)=60$
50	1/50	$100\cdot 1/50=2$	$1500/50=1500\cdot (2/100)=30$