• Understanding how subtraction relates to addition
• Keeping a clear head when subtraction looks complicated
• Mastering quick methods to do in your head
As darkness is to light, and sour is to sweet, so subtraction is to addition. As we shall see in this chapter, this relationship between adding and subtracting is useful for understanding and calculating subtraction-based problems. If you have 7 carrots, and you add 3, and then you take away 3, you are left exactly where you started, with 7. So subtraction and addition really do cancel each other out.
Subtraction is also known as taking away, for good reason. If you have 17 cats, of which 9 are Siamese, then the number of non-Siamese cats is given by taking away the number of Siamese from the total number, that is, by subtracting 9 from 17.
Now, there is one important theoretical way that subtraction differs from addition: when we calculate 17 + 26, the answer is the same as for 26 + 17. Swapping the order of the numbers does not make any difference to the answer. But, with subtraction, this is no longer true: 26 − 17 is not the same as 17 − 26. In a later chapter we will look at the concept of negative numbers which give meaning to expressions such as 17 − 26. In this chapter, we will stick to the more familiar terrain of taking smaller numbers away from larger ones. (As it happens, extending these ideas into the world of negative numbers is simple: while 26 − 17 is 9, reversing the order gives 17 − 26, which comes out as −9. It is just a matter of changing the sign of the answer. But we shall steer clear of this for the rest of this chapter.)
The techniques for subtraction mirror the techniques for addition, with just a little adjustment needed. And, as with addition, the first step is to get comfortable subtracting small numbers in your head.
HAVE A GO AT QUIZ 1.
As ever, if you feel you could do with more practice, then set yourself your own challenges in batches of five, starting as slowly as you like, and aiming to build up speed and confidence gradually.
Now we move on to numbers which are more than just one digit long. These larger calculations can be set up in a very similar way to addition as this chapter’s golden rule tells us.
The first thing to do is to align the two columns one above the other, making sure that units are aligned with units, tens with tens and so on. Then the basic idea is just to subtract the lower number in each column from the upper number. So to calculate 35 − 21 we would write this:
EASY? THEN PRACTICE BY DOING QUIZ 2!
What can go wrong with the procedure in the last section? Well, we might face a situation like this:
The first step is to attack the units column. But this seems to require taking 7 from 6, which cannot be done (at least not without venturing into negative numbers, which we are avoiding in this chapter). So what happens next? When we were adding, we had to carry digits between columns. In subtraction, the opposite of carrying is borrowing. It works like this: we may not be able to take 7 from 6, but we can certainly take 7 from 16. The way forward, therefore, is to rewrite the same problem like this:
Notice that the new top row “forty-sixteen” is just a different way of writing the old top row “fifty-six.” With this done, the old procedure of working out each column individually, starting with the units, works exactly as before.
What went on in that rewriting of the top row? We want to speed the process up. Essentially, one ten was “borrowed” from the tens column (reducing the 5 there to 4) and moved to the units column, to change the 6 there to 16. Usually, when writing out these sort of calculations, we would not bother to write a little 1 changing the six to sixteen, since this can be done in your head. But if it helps you to pencil in the extra 1, then do it! It is usual, however, to change the 5 to 4 in the tens column. To take another example, if we are faced with 94 − 36, the way to write it out is like this:
WHAT’S GOING ON HERE? TEST YOURSELF WITH QUIZ 3.
This column-based method is very reliable and efficient. But, just as we saw in the case of addition, it is not ideal when you want to calculate in your head, instead of on paper. The first purely mental technique we looked at for adding was splitting numbers up: to add 32 to 75, we split 32 up into 30 and 2, and then added these on separately, first 75 + 30 = 105, and then 105 + 2 = 107.
This approach works just as well with subtraction. (You might want to remind yourself of how it worked for adding before continuing.)
TRY QUIZ 4. CAN YOU WORK IT OUT IN YOUR HEAD?
In the context of subtraction, it is always the number being taken away that gets split up. Suppose I know that there are 75 people in my office, of whom 32 are men. I want to know how many women there are. The calculation we need to work out is 75 − 32. The technique again involves splitting the 32 up into 30 and 2. So first we take away 30 from 75, to get 45, and then subtract the final 2, to leave the final answer of 43 women. The aim is to complete the subtraction by splitting the numbers up, without writing anything down. But, for practice, you might want to write down the intermediate step, that is, 45 in the above example.
Another mental trick we learned for adding was rounding up and cutting down. This works just as well for subtraction. The only thing to watch out for is whether the numbers should be going up or down.
GIVE IT A GO YOURSELF WITH THE FINAL QUIZ, NUMBER 5.
For example, to calculate 80 − 29, it might be convenient to round 29 up to 30. This gives us 50. It is in the final step that we need to take care. Instead of cutting the answer down by 1 (as we did when adding), this time we have subtracted 1 too many. So we have to add 1 back on, to arrive at a final answer of 51.
Sum up Subtraction is the opposite of addition. Once you know how to do one, it is just as easy to do the other!
1 Getting started
a 9 − 6
b 8 − 4
c 12 − 5
d 17 − 9
e 16 − 8
2 Write in columns and subtract.
a 54 − 33
b 89 − 61
c 748 − 318
d 6,849 − 4,011
e 19,862 − 17,722
3 Get borrowing!
a 72 − 18
b 56 − 39
c 81 − 47
d 178 − 159
e 218 − 119
4 Split these up, to work out in your head.
a 60 − 23
b 75 − 14
c 54 − 32
d 73 − 24
e 101 − 43
5 Work out in your head, rounding up and adding on.
a 67 − 29
b 73 − 18
c 64 − 38
d 87 − 49
e 110 − 68
