Long Division Shortcut (Part 1)

(Is that title an oxymoron?)

Imagine you have to do this division:

926/18

How would you do it? Would you rewrite it with that funny division symbol (“division bracket,” or “right parenthesis followed by a vinculum over the dividend”)? Would you use a calculator? (Please say “no” to that!)

After you rewrote it, would you start by trying to figure out how many times 18 would go into 92? If you did, you would be doing it the way most people learned in school, and you would be wasting a lot of time and effort.

Look at it again. Your school probably spent a lot of time trying to teach you about factors when you were young. But it probably didn’t take too well, because they probably didn’t teach you at all – they just did mathematical show-and-tell over and over, and called it “teaching.”

Have no fear, your brain is capable of much, much more. Let’s start…

You’ll probably immediately notice that both numerator and denominator are even numbers.

Using factors, we can see that 2 goes into both numbers. Now we’re turning the problem into short division. Without rewriting anything, we can simply cut 926 in half, from left to right, getting 463. Do the same with 18 and get 9.

You’ve successfully turned the problem into 463/9. PLEASE don’t write it down. You can keep a number in your head like that if you practice. Start practicing now. You will be happy you did.

Next we do short division again. 9 goes into 46 five times, with a remainder of 1. The remainder goes in front of the 3 (of the 463) giving us 13. 9 goes into 13 one time, with a remainder of 4. It would seem that the answer is fifty-one, remainder 4.

There is just one other thing to remember. Since you reduced the original problem by a factor of 2, even though you kept the proportion of the improper fraction (that means that the improper fraction 926/18 is the same thing as 462/9) the remainder is not the same. You must factor the 2 back into remainder. 2 x 4 = 8, so your real answer would be fifty-one, remainder 8.

Make sure you remember this:
If you factor a division problem before you solve it, you must multiply the remainder by that factor after you are done.

Had you not kept the remainder, and instead, continued to divide the problem until you either reached a finite decimal number, or a repeating decimal, then you wouldn’t have to worry about “unfactorizing” the remainder.

What? You’d want to check it? Well, you could multiply 51 by 18, add the remainder of 8 and get 926, or you could multiply 51 by 9, add the remainder of 4 and get 463. Or you could even do the original long division.

But, you, you clever custard you, you read the previous post, right? And you know about checking by crunching. Or didn’t you? Well, it’s never too late…

In the next post about long division shortcuts, we’ll talk a little more about factoring division problems.