Subtraction facts and strategies
Subtraction is
NOT commutative, which means that the order matters. It is important to realise that:
5 - 3 is not the same thing as
3 - 5 .
Methods of *mental* subtraction
Method 1 : Make addition easier by rounding
up the right-hand number to a multiple of 10.
The important thing is that when modifying subtractions in this way, you must always
take away or add the same number from both numbers - this is different to addition, where you must always do the opposite to each number.
So for example, we can make
26 - 17 easier by adding 3 to the 17 and the 26, with the goal of turning the 17 into 20, a nice round numbers. So the calculation becomes:
29 - 20 which comes to
9, which is a lot easier than the previous version.
You can continue the process to round up to multiples of 100, for larger subtractions. For exmple,
113 - 26
- First adjust it by adding 4 to both sides:
117 - 30 . Now, add 70 to both sides:
187 - 100 - now we can see straight away that this comes to 87.
As a tip, this makes 'column subtraction' with paper and pencil much easier as well. So for example, with 113 - 26, as above, this involves two ugly, error-prone 'borrows' from a more significant digit. But:
187 -
100
---
87
Is much easier, simply because you are mainly doing a subtraction of zero for most of the digits.
Method 2 : This is the way the 'super mental calculators' do it. Basically, unlike how you were taught at school, you subtract from left to right, not right to left. This better matches the way we say the number aloud, and actually, it's much quicker than all other methods when you get used to it.
It relies on two things:
- A 'carry' is never more than 1
- You need to be able to count down ( by subtracting ) single digits *instantly*.
There are only two scenarios for two digits:
- they subtract without going 'bust' e.g. 6 - 3 = 3
- they subtract and go bust, e.g. 5 - 7 = -2.
In the second case, don't think of this as '-2,' instead, the answer is 8 and a *carry.*
The '8' is the 'complement' of -2 : If we had to count up -10 to -2, it would be eight steps.
This is the skill you have to develop.
By far the easiest way to do this at the start is to add ten to the left-hand number ( the 'minuend' .) You only
do this if this number is less than the second ( the 'subtrahend. ) So for example:
5 - 7 : Think of this as 15 - 7 = 8, plus a 'subtraction carry.'
2 - 8 : Think of this as 12 - 8 = 4
3 - 9 : Think of this as 13 - 9 = 4 , and so on.
Worked example ( easy ):
1 4 5 +
7 2
Step 1) 1 minus nothing is 1. But look next door: 4 - 7 goes bust, so subtract the carry ( 1 ) from our current answer : 1 - 1 = 0
Step 2) 4 - 7 = -3 so answer digit is 7. But! look next door: 5 + 2 does not go bust, answer digit stays as 7.
Step 3) 5 - 2 = 3. Answer is 73.
Harder example:
1 0 0 4 4 -
9 2 7 3
Step 1) Look at 0 and 9, this goes bust so ignore the left-most 1 at the top
( bit quicker if you can learn to do this )
Step 2) 0 - 9 = -1 so answer digit is 1. But look next door: 0 - 2 goes bust, so our answer is still 0 as 1 - 1 = 0
Step 3) 0 - 2 = -2 so answer digit is 8. But look next door: 4 - 7 goes bust so subtract 1 from answer digit: 7
Our answer 'so far' is 7
Step 4) 4 - 7 = -3 so answer digit is 7. But look next door: 4 - 3 does not go bust so answer digit remains at 7.
Our answer so far is 77.
Step 5) 4 - 3 = 1 . Answer is 771.
Check: 9273 +
771
10044
Note!
As with normal column subtraction, this method does not work if the answer is going to be negative!
Hence, say you need to work out 17 - 35, you need to work out 35 - 17 , and then just
stick a minus on the answer:
Right:
3 5 - : 3 minus 1 is 2, but the next column goes bust so make this a 1.
1 7 15 minus 7 is 8. Answer is 18, so overall adjusted answer is -18.
Wrong:
1 7 - : 1 minus 3 is -2. 17 - 5 is 12, which is a positive carry, answer is -12.
3 5 Doesn't work!
You can however do something called *signed* column subtraction in order to make it work,
but this is fiddly and fraught with the potentian for mistakes, so don't use it.
It's just here for example:
Right:
1 7 - : 1 minus 3 is -2. 7 - 5 is 2. Now combine!
3 5 (- 2 x 10 ) + 2 --> -20 + 2 = -18.
Another example:
1 2 - : 1 minus 4 is -3, which really means (-3 x 10 ) = -30 . -33
4 5 2 minus 5 is -3. Combine: -30 + -3 = -33
** A much better way : **
The principle of 'making the subtrahend nice and round' also helps for negative answers,
but this time you make the *top* number nice and round ( the minuend: )
1 2 - : 12 is 'rubbish' so add 8 to top and bottom: 2 0 -
4 5 5 3
.. now do 53 - 20 = 33 , and bung a minus on to give: -33.
This is usually the fastest and easiest way to do things.
"Nines complement"
As a mental challenge, you can do subtractions using "nines complement" which is the way
computers do it - except they use "twos complement" as they work in binary.
Example:
1 2 - Do 99 - 45 to give 54.
4 5 Add on the 12: 66.
Subtract 66 from 99: 99 - 66 = 33.
Bung a minus on: -33.
Nines complement works with any subtraction, whether the answer is positive or negative.
THe advantage of this method is that you only ever have to do "non carry" subtractions.
Difficulty Levels only apply to the basic mental maths section. typical questions :
1: 19 - 6
2: 61 - 15
3: 315 - 49
The numbers may seem big for the 'basic' section, but the difference is that the advanced subtraction option
guarantees to make it a difficult subtraction where a carry is needed on every digit except the most significant.