Long Division

Background

When we have easy to divide numbers, say like , it's easy to see what the answer is (= 2). Is there a way to work out a much bigger division without using a calculator?

Question

Evaluate:

1. 
2.  using a remainder
3.  using a fractional remainder

Answer

There is a way - long division.

1.  
2.  
3.  

Analysis

While the art of long division is less important nowadays than it was before the advent of calculators, the ability to do long division will help when dividing terms that are more complicated.

So let's go over how to do this. (The link below to www.mathisfun.com has an excellent description on how to do long division.)

Let's first look at the Question 1: . Before we dive in, notice that we can also express the question as . 3836 is the numerator and 7 is the denominator.

We set up the long division by putting the numerator under a "division sign" and we put the denominator outside of that sign:



Now we're going to multiply 7 by a natural number so that we get a product bigger than the first digit of the numerator. And as we can see, there is no such number because 7 is bigger than 3.

So now what we do is do the same thing, multiply 7 by a natural number, but this time we look at the first two digits of 3836, which is 38. We want to get as close as possible but not go over. The closest we can get is 5: 7 x 5 = 35 but 7 x 6 = 42. So our quotient has as a first digit 5 (normally the line continues across over the numerator, but I can't get my equation editor to do it, so it's an over brace here):



Ok, we know 7 x 5 = 35 (we did that just above). We subtract 35 from 38 (the first two digits of the numerator) and get a difference of 3. We now make a new number for the 7 to divide into - we take the difference we just made (the 3) and take the next number of the numerator (also a 3 here) - and so we have 33 for the 7 to divide into (again, the mathisfun.com link has the more traditional look in its presentation):



We again find the largest natural number we can to multiply against the 7 so that we get as close as possible to 33 without going over. That number is 4: 7 x 4 = 28 but 7 x 5 = 35:



Our answer now has 5 and 4 in it.

4 x 7 = 28. 33 - 28 = 5 so we put the 5 in front of the 6:



Let's find the largest multiple of 7 we can make that gets as close to 56 as we can get without going over. It turns out in this case that 7 x 8 = 56 - perfect! 7 divides into 3836 evenly:



And so 3836 / 7 = 548

Question 2

Let's do something smaller to show how to work with a remainder. 

We could use the whole "dividing bar" thing and put the 3 on one side and the 16 under it. But I think we can look at this and see that 3 doesn't divide equally into 16: 3 x 5 = 15 and 3 x 6 = 18.

So our answer is 5... and something. Let's work with what that something is.

3 x 5 = 15. If we subtract 15 from 16, we have 16 - 15 = 1. What to do with that 1.

One way to deal with it is to say it's the remainder. What does that mean?

Let's think of it this way - if we have 16 freshly baked chocolate chip cookies and there are 3 people who are going to share them equally, what can we do? Each person can get 5 cookies, and so that's 15 cookies eaten. That last cookie gets put into the cookie jar for later. That last cookie is the remainder.

Question 3

So what if the three people don't want to put the cookie into the jar and instead want to share it too? Well... we can cut the cookie, can't we? There are three people, so we cut the cookie into three pieces ("thirds") and each person gets one piece each.

Each person then gets 5 whole cookies and 1 piece out of 3 pieces of that last cookie. That's 5 plus 1/3 cookies for each person. We can express this as:



Vocabulary used:

For more information check out these links (comment to add your favourite link):

https://www.mathsisfun.com/long_division.html

Where might you have come from?

Fact-orials Index

Operations:

• Division

Associated Operations:

• Dividing Evenly
• Remainders and Modulo

Where might we go?