Mental Math - The Heads and Tails Multiplication Method

Problem: Calculate the value of the expression

        103 x 107 – 91 x 99

The tail of a number is defined to be the last digit of the number. The head of the number is defined to be all of the digits except last digit. If you want to multiply two numbers, whose heads are the same and whose tails add up to 10, then you can use the Chinese Head and Tail method to multiply the two numbers.

Let's take a look at multiplying 91 times 99. These two numbers have the same head, “9”. The tails of the two numbers add up to 10 (1 + 9 = 10).

First, we want to multiply the head times the head plus one. It this example we multiply 9 times 9 + 1 (or 10). We get 90 as a result.

Next we look at the tails of the two numbers. In the example, the two tails are 1 and 9. We multiply 1 x 9 and get 9. Since we need two digits rather than just one, we will use 09.

Finally, we write the digits that we got from the heads followed by the digits that we got from the tails. That is, 9009. So the product of 91 and 99 is 9009.

Let try one more example. Let’s calculate 103 x 107. Here the heads are the same (both are 10) and the tails 3 and 7 add up to ten. So our head and tail method should allow us to multiply these two numbers. First, we multiply 10 times 10 + 1, and we get 110. Next we multiply 3 times 7 and get 21. Now we simply attach these two numbers and get 11021, which is the product of 103 and 107.

The final step in calculating the value of the expression 103 x 107 – 91 x 99 is to subtract the products that we just calculated. We get 11021 – 9009 = 2012.