Chapter 17
CHECKING ANSWERS (DIVISION)
Casting out nines is one of the most useful tools available for working with mathematics. I use it almost
every day. Casting out nines is easy to use for addition and multiplication. Now we are going to look at
how we use the method to check division calculations.
Changing to multiplication
When we looked at casting out nines to check a problem with subtraction, we found we often had to
reverse the problem to addition. With division, we need to reverse the problem to one of multiplication.
How do we do that?
Let’s say we divide 24 by 6 to get an answer of 4. The reverse of that would be to multiply the answer
by the divisor to get the original number we divided. That is, the reverse of 24 ÷ 6 = 4 is 4 × 6 = 24.
That is not difficult.
To check the answer to the problem 578 ÷ 17 = 34, we would use substitute numbers.
Substituting, we have 2 ÷ 8 = 7. That doesn’t make sense. So, we do the calculation in reverse. We
multiply: 7 × 8 = 2. Does 7 times 8 equal 2 with our substitute numbers?
7 × 8 = 56
5 + 6 = 11
1 + 1 = 2
Seven times 8 does equal 2; our answer is correct.Handling remainders
How would we handle 581 ÷ 17 = 34, with 3 remainder? We would subtract the remainder from 581 to
make the calculation correct without a remainder. We could either subtract the remainder first to get 578
÷ 17 = 34, which is the problem we already checked, or we do it in the casting out nines, like this:
(581 − 3) + 17 = 34
Reversed, this becomes 17 × 34 = 581 − 3.
Writing in the substitutes we get:Or, 8 × 7 = 5 − 3.
8 × 7 = 56
5 + 6 = 2
5 − 3 = 2
The answer checks correctly.
If you are not sure how to check a problem in division, try a simple problem like 14 ÷ 4 = 3 r2.
Reverse the problem, subtracting the 2 remainder from 14 to get 3 × 4 = 14 − 2. Then apply the