Chapter 5
CHECKING YOUR ANSWERS
What would it be like if you always found the right answer to every maths problem? Imagine scoring
100% for every maths test. How would you like to get a reputation for never making a mistake? If you
do make a mistake, I can teach you how to find and correct it before anyone (including your teacher)
knows anything about it.
When I was young, I often made mistakes in my calculations. I knew how to do the problems, but I
still got the wrong answer. I would forget to carry a number, or find the right answer but write down
something different, and who knows what other mistakes I would make.
I had some simple methods for checking answers I had devised myself but they weren’t very good.
They would confirm maybe the last digit of the answer or they would show me the answer I got was at
least close to the real answer. I wish I had known then the method I am going to show you now.
Everyone would have thought I was a genius if I had known this.
Mathematicians have known this method of checking answers for about 1,000 years, although I have
made a small change I haven’t seen anywhere else. It is called the digit sum method. I have taught this
method of checking answers in my other books, but this time I am going to teach it differently. This
method of checking your answers will work for almost any calculation. Because I still make mistakes
occasionally, I always check my answers. Here is the method I use.
Substitute Numbers
To check the answer to a calculation, we use substitute numbers instead of the original numbers we
were working with. A substitute on a football team or a basketball team is somebody who takes another
person’s place on the team. If somebody gets injured, or tired, they take that person off and bring on a
substitute player. A substitute teacher fills in when your regular teacher is unable to teach you. We can
use substitute numbers in place of the original numbers to check our work. The substitute numbers are
always low and easy to work with.
Let me show you how it works. Let us say we have just calculated 12 × 14 and come to an answer of
- We want to check this answer.
12 × 14 = 168
The first number in our problem is 12. We add its digits together to get the substitute:
1 + 2 = 3
Three is our substitute for 12. I write 3 in pencil either above or below the 12, wherever there is
room.
The next number we are working with is 14. We add its digits:
1 + 4 = 5
Five is our substitute for 14.
We now do the same calculation (multiplication) using the substitute numbers instead of the original
numbers:
3 × 5 = 15
Fifteen is a two-digit number so we add its digits together to get our check answer: