7 Division

You know from the last section that multiplication is repeated addition, similarly division can be thought of as repeated subtraction. For example, 20 divided by 4 is 5 (20 ÷ 4 = 5). The answer is 5 because 5 is the number of times you subtract 4 from 20 to arrive at zero:

Figure 16 Demonstrating division

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This shows 5 being subtracted from 20 five times. Giving, 20 − 4 = 16, 16 − 4 = 12, 12 − 4 = 8, 8 − 4 = 4 and 4 − 4 = 0.

Figure 16 Demonstrating division

This also leads to the realisation that division is the opposite or undoes multiplication. If you add 4, five times to zero (repeated addition or multiplication) then you will arrive back at 20.

And just as with the other operators, there are words that tell us you need to divide.

Here are some examples that you may recognise:

• How many times does 8 go into 72?
• How can you share 72 items equally among eight people?
• How many lots of five are there in 20?

Unlike multiplication when there is only one symbol that says multiply, division has a whole family of notation! If you want to divide 72 by 8 this can be written in the following ways:

72 ÷ 8, or , or 72/8 or

Not all division problems will result in a nice whole number answer and you may have a number left over. This number is known as the remainder. The common notation is to show this using the letter R.

Now, you’re going to look at division on paper, just as you did with multiplication. You set up division on paper using the final notation from the list above. Starting from the left divide into each number in turn, writing the result above the number you have divided into. Watch the video below to see how this works.

Download this video clip.Video player: swm_1_s1_2568_divided_3.mp4

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SPEAKER 1:

First, let's look at doing division by dividing by quite a small number. Let's do 2,568 divided by 3. This is how it is written. The number you are dividing by goes at the start. That's the 3. And the number you are dividing by 3 goes next that's the 2,568. The first step is to look at the first digit of 2,568, the two, and say how many 3's go into two.

Well, 3's don't go into two. So you should carry the 2 over and say how many 3's go into 25. The answer to that is 8, as 3 times 8 is 24. You show that by writing an eight above the top line above the five. But you've also got one left over as eight times 3 is 24 not 25. So you need to carry over that one. Now you need to ask yourself, how many 3's are there in 16? The answer is 5, as three times 5 is 15.

But again, there is one left over. So you put a 5 above the top line, above the 6 and carry the 1. Next, you need to ask yourself how many 3's are there in 18. The answer is 6 and nothing left over. So put a 6 above the 8 to give 856.

You can also check your answer by working out three times 856. If you get the answer, 2,568, that shows the calculation is correct.

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Division also works in a similar way when you are dividing a number involving decimals by a whole number. In cases like this, you just have to be careful to put the decimal point in the right place.

Wacth the video below to see an example.

Download this video clip.Video player: swm_1_s1_23_6_divided_5.mp4

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SPEAKER 1:

To divide 23.6 by 5, you would first ask yourself, how many 5s in two? As 5 doesn't go into two, you would carry the two, and ask yourself, how many 5s in 23? Since 4 times 5 is 20 the answer is 4 with 3 left over that need to be carried. So now you need to add the 4 above the top line above the 3. Here you can also see the 3 has carried over to be with the 6 after the decimal point. Even though this is after the decimal point, the carrying just works like before.

At this point, before you do anything else you need to put a decimal point on the top line exactly above the one under the line. Now, you should ask yourself, how many 5s in 36? The answer is 7 as 7 times 5 is 35, with 1 left over that you'll need to carry. So put the 7 on the top line after the decimal point.

You may now be thinking, how am I going to carry the 1 as there aren't any more digits left? The way to do this is to put an extra 0 after the 6. This is OK to do is 23.6 is the same number as 23.60. The 0 just means there is zero hundredths. Now, ask yourself, how many fives in 10? The answer to that is 2, and nothing left over. So you'd put the 2 above the 0, and you have finished. The answer to 23.6 divided by 5 is 4.72.

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What happens if you have to divide by a larger number – something like 16, say? It’s easiest to do it using your calculator, but if you don’t have your calculator, you can still do it using this method. The next video shows how.

Download this video clip.Video player: swm_1_s1_632_16.mp4

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SPEAKER 1:

Suppose you wanted to do 632 divided by 16? Before you start, unless you know your 16 times table, which most people don't, it can help to jot down the multiples of 16. You can work these out by adding 16 to the previous one. You'll see how to use them as we go through.

As normal, start by asking yourself, how many 16s go into 6? As 16 doesn't go into 6, you need to carry the 6 over. Now ask yourself, how many 16s are there in 63? This is where the list of multiples of 16 comes in handy.

You can see that 3 times 16 is 48, and 4 times 16 is 64. So only 3 16s go into 63, with 15 left over as 63 minus 48 is 15. So as normal, you should put the 3 above the line and carry over 15.

The next step is to ask, how many 16s in 152? The list tells you that 9 times 16 is 144, so the answer is 9, with 8 left over. So you put a 9 above the line, but you also need to carry that 8 to somewhere. Here, what you need to do is put a decimal point after the 2 in 632, and a 0 after it. This is fine to do, as 632.0 is the same number as 632. It just means 632 and 0 tenths.

That now gives you somewhere to carry the 8. Before doing any more division, you need to remember to put a decimal point above the line, then ask yourself, how many 16s in 80? And the answer from the list is 5, as 5 times 16 is 80. So 632 divided by 16 is 39.5.

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Now have a go at the next activity. If you wish you can view the times table [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)]   again.

When looking at subtraction and addition, you thought about whether it mattered in what order these were carried out. You found that for addition the answer is the same whatever the order, but that this is not the case for subtraction. What about multiplication and division? You’ll find out in the next section.