There is another shortcut to this procedure. If we find a 9 anywhere in the calculation, we cross it out.
This is called casting out nines. You can see with this example how this removes a step from our
calculations without affecting the result. With the last answer, 196, instead of adding 1 + 9 + 6, which
equals 16, and then adding 1 + 6, which equals 7, we could cross out the 9 and just add 1 and 6, which
also equals 7. This makes no difference to the answer, but it saves some time and effort, and I am in
favour of anything that saves time and effort.
What about the answer to the first problem we solved, 168? Can we use this shortcut? There isn’t a 9
in 168.
We added 1 + 6 + 8 to get 15, then added 1 + 5 to get our final check answer of 6. In 168, we have
two digits that add up to 9, the 1 and the 8. Cross them out and you just have the 6 left. No more work
to do at all, so our shortcut works.
Check any size number
What makes this method so easy to use is that it changes any size number into a single-digit number.
You can check calculations that are too big to go into your calculator by casting out nines.
For instance, if we wanted to check 12,345,678 × 89,045 = 1, 099,320,897,510, we would have a
problem because most calculators can’t handle the number of digits in the answer, so most would show
the first digits of the answer with an error sign.
The easy way to check the answer is to cast out the nines. Let’s try it.
All of the digits in the answer cancel. The nines automatically cancel, then we have 1 + 8, 2 + 7, then
3 + 5 + 1 = 9, which cancels again. And 0 × 8 = 0, so our answer seems to be correct.
Let’s try it again.
137 × 456 = 62,472
To find our substitute for 137:
1 + 3 + 7 = 11
1 + 1 = 2
There were no shortcuts with the first number. Two is our substitute for 137.
To find our substitute for 456:
4 + 5 + 6 =
We immediately see that 4 + 5 = 9, so we cross out the 4 and the 5. That just leaves us with 6, our
substitute for 456.
Can we find any nines, or digits adding up to 9, in the answer? Yes, 7 + 2 = 9, so we cross out the 7
and the 2. We add the other digits:
6 + 2 + 4 = 12
1 + 2 = 3
Three is our substitute answer.
I write the substitute numbers in pencil above or below the actual numbers in the problem. It might
look like this:
Is 62,472 the right answer?
We multiply the substitute numbers: 2 times 6 equals 12. The digits in 12 add up to 3 (1 + 2 = 3). This