method to the problem that you want to check.
Finding the remainder with a calculator
When you carry out a division with your calculator, it gives your remainder as a decimal. Is there an
easy way to find out the true remainder instead?
Yes, there is. If you divide 326 by 7 with a calculator, you get an answer of 46.571428 with an eight-
digit calculator. What if you are trying to calculate items you have to divide up; how do you know how
many will be left over?
The simple way is to subtract the whole number before the decimal and just get the decimal part of
the answer. In this case you would just subtract 46, which gives 0.571428. Now multiply this number by
the number you divided by. For our calculation, we multiply 0.571428 by 7 to get an answer of
3.999996. You round the answer off to 4 remainder.
Why didn’t the calculator just say 4? Because it only works with numbers to a limited number of
decimal places, so the answer is never exact.
Multiplying the decimal remainder will almost always give an answer that is fractionally below the
correct remainder. You will be able to see this for yourself. If you are dividing 326 chairs into 7
classrooms at school, you know you won’t have 46 in each room with 3.999996 chairs left over. You
would have 4 chairs to keep for spares.
Bonus: Casting twos, tens and fives
Just as it is possible to check a calculation by casting out the nines, you can cast out any number to
make your check.
Casting out twos will only tell you if your answer should be odd or even. When you cast out twos, the
only substitutes possible are 0 when the number is even and 1 when the number is odd. That is not very
helpful.
When I was in primary school I sometimes checked answers by casting out the tens. All that did was
check if the units digit of the answer was correct. Casting out the tens means you ignore every digit of a
number except the units digit. Again, it is not very useful, but I did use it for multi-choice tests where a
check of the units digit was sometimes enough to recognise the correct answer without doing the whole
calculation.
Let’s try an example. Which of these is correct?
34 × 72 =
a) 2,345
b) 2,448
c) 2,673
d) 2,254
Multiplying two even numbers can’t give an odd number for an answer. (That is casting out twos.)
That eliminates a) and c). To check the other two answers, we multiply the units digits of our problem
together and get an answer of 8 (4 × 2 = 8). The answer must end with 8, so the answer must be b).
Casting out fives is another option. For the above calculation, the substitute numbers would be
exactly the same. But it can have its uses for checking multiplication by small numbers.
Let’s try 7 × 8 = 56 as an example.
Casting out fives we get 2 × 3 = 1.
We divide by 5 and just use the remainder. Again, we are only working with the units digits. Five
divides once into 7 with 2 remainder; it divides once into 8 with 3 remainder and, just using the 6 of 56,