is the same as our substitute answer so we were right again.
Let’s try one more example. Let’s check if this answer is correct:
456 × 831 = 368,936
We write in our substitute numbers:
That was easy because we cast out (or crossed out) 4 and 5 from the first number, leaving 6. We cast
out 8 and 1 from the second number, leaving 3. And almost every digit was cast out of the answer, 3
plus 6 twice, and a 9, leaving a substitute answer of 8.
We now see if the substitutes work out correctly: 6 times 3 is 18, which adds up to 9, which also gets
cast out, leaving 0. But our substitute answer is 8, so we have made a mistake somewhere.
When we calculate it again, we get 378,936.
Did we get it right this time? The 936 cancels out, so we add 3 + 7 + 8, which equals 18, and 1 + 8
adds up to 9, which cancels, leaving 0.
This is the same as our check answer, so this time we have it right.
Does this method prove we have the right answer? No, but we can be almost certain.
This method won’t find all mistakes. For instance, say we had 3,789,360 for our last answer; by
mistake we put a 0 on the end. The final 0 wouldn’t affect our check by casting out nines and we
wouldn’t know we had made a mistake. When it showed we had made a mistake, though, the check
definitely proved we had the wrong answer. It is a simple, fast check that will find most mistakes, and
should get you 100% scores in most of your maths tests.
Do you get the idea? If you are unsure about using this method to check your answers, we will be
using the method throughout the book so you will soon become familiar with it. Try it on your
calculations at school and at home.
Why does the method work?
You will be much more successful using a new method when you know not only that it does work, but
you understand why it works as well.
Firstly, 10 is 1 times 9 with 1 remainder. Twenty is 2 nines with 2 remainder. Twenty-two would be 2
nines with 2 remainder for the 20 plus 2 more for the units digit.
If you have 35¢ in your pocket and you want to buy as many lollies as you can for 9¢ each, each 10¢
will buy you one lolly with 1¢ change. So, 30¢ will buy you three lollies with 3¢ change, plus the extra
5¢ in your pocket gives you 8¢. So, the number of tens plus the units digit gives you the nines
remainder.
Secondly, think of a number and multiply it by 9. What is 4 × 9? The answer is 36. Add the digits in
the answer together,3 + 6, and you get 9.
Let’s try another number. Three nines are 27. Add the digits of the answer together, 2 + 7, and you get
9 again.
Eleven nines are 99. Nine plus 9 equals 18. Wrong answer? No, not yet. Eighteen is a two-digit
number so we add its digits together: 1 + 8. Again, the answer is 9.
If you multiply any number by 9, the sum of the digits in the answer will always add up to 9 if you
keep adding the digits until you get a one-digit number. This is an easy way to tell if a number is evenly
divisible by 9. If the digits of any number add up to 9, or a multiple of 9, then the number itself is
evenly divisible by 9.
If the digits of a number add up to any number other than 9, this other number is the remainder you
would get after dividing the number by 9.