In school, the teacher is always on top of you for checking your work. When you do a subtraction problem, solve the reversed addition problem and make sure it is right, when you do an algebra problem, make sure you plug your solution back into the original equation. These are all things that are drilled into our heads, but never quite executed.
I did post a year and a half ago about checking your work in algebra problems: plugging the answer into the original equation (click here to see how to do that). But there is also a shortcut for checking work on plain arithmetic problems as well.
Let's take the problem 138 + 253. I would have went smaller, but the method will be easier to demonstrate with larger numbers.
If we add that up normally, we would get:
How do we know if that is correct? Well, we do something called mod sums. What that means is we add up the digits in the number, and then add up the digits in this sum, and keep going until we find a single digit number. This is called the number's mod sum or digital root.
So, what is the mod sum of 138? Well, we add up the digits.
1 + 3 + 8 = 12
1 + 2 = 3
So, the mod sum or digital root of 138 is 3. Let's find it for 253.
2 + 5 + 3 = 10
1 + 0 = 1
The mod sum of 253 is therefore 1. Let's find the mod sum of the total and see if you notice the pattern.
3 + 9 + 1 = 13
1 + 3 = 4
So, the two addends have mod sums of 3 and 1. The sum has a mod sum of 4. What is the pattern? That's right, the mod sum of the answer is the sum of the mod sums of the addends. What about a subtraction problem?
The answer to this problem is 281. But how do we confirm it?
The mod sum of 924 is 6 (9+2+4=15 and 1+5=6) and the mod sum of 643 is 4 (6+4+3=13 and 1+3=4). So, the mod sum of the difference must be the difference of the two mod sums. The mod sum of 281 is 2 (2+8+1=11 and 1+1=2), which is the difference of 6 and 4. So, the answer was correct.
What about a multiplication problem? Say 71 x 55. If you do the math, you will find that the answer is 3905. But let's check it with mod sums.
Mod Sum of 71 = 8
Mod Sum of 55 = 1
Mod Sum of 3905 = 8
8 x 1 = 8
So it is correct. There are some glitches in the technique, but this is the basis of it. You might run into scenarios that I didn't quite explain how to deal with, but feel free to comment. I will be happy to respond with some more specific pointers. Have fun actually checking your work now!