Tricks with Arithmetic
What is 26 × 34? What about 37 × 13?
There are some neat little tricks you can remember which will help you do these and other calculations in your head in seconds. Here we give examples and explain why they work.
The Difference of Squares
Multiplying 26 by 34 might not seem to have anything to do with square numbers, but look at it like this:
| 26 × 34 | = (30 – 4)(30 + 4) |
| = 30² + (30 × 4) – (4 × 30) – 4² |
This method works in general; for two numbers a and x,
(a – x)(a + x) = a² – x².
So, to give another example,
17 × 23 = 20² – 3² = 391.
Know Numbers
My sister once said “Numbers are my friends.” She’s never been able to live it down, of course, but she may have had a point. If you get to know some interesting habits of a few numbers, mental arithmetic certainly becomes a lot simpler.
For instance, it’s very useful to know that 17 × 3 = 51, and not only because it reminds you that 51 isn’t prime.
For example:
| 24 × 17 | = (8 × 3) × 17 | |
| = 8 × (3 × 17) | = 8 × 51 | |
| = 408. |
To return to the second problem at the beginning of the article, the number 37 is interesting because
| 37 × 3 | = 111. To write this more usefully, |
| 37 × 3 | = (100 × 1) + (10 × 1) + (1 × 1). So, |
| 37 × 3a | = a ((100 × 1) + (10 × 1) + (1 × 1)) |
| = 100a + 10a + a . |
This means that 37 × 12 = 400 + 40 + 4 = 444, and so 37 × 13 = 444 + 37, which is 481.
Now try 37 × 42.
General points
- Try factorising the numbers involved; 512 might be a lot easier to deal with if you remember it’s a power of 2.
- Check your short cuts. 37 × 12 can’t be much different from 40 × 10 = 400, so if you get something wildly different, there’s been a slip at some point.
- Keep practising!
Sarah Perkins
