### Mental Math - Tails of 5 Multiplication Method

Problem: Calculate the value of the expression

45 x 65

Method: We can use the Tails of 5 method to help solve this problem.

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 tails are both 5, then you can use the Tails of 5 method to multiply the two numbers quickly.

Let's take a look at multiplying 45 times 65. These two numbers both have a tail of 5. We know that the product of these two number will end in 25 (or possibly 75). The other digits of the product can be determined by multplying the heads of the two numbers and then adding the average of the two heads. In this case the two heads are 4 and 6. When we multiply 4 times 6 we get 24. The average of 4 and 6 is 5 so we add the 5 to 24 and get 29. So our product starts with 29 and ends with 25. Therefore, the product of 45 x 65 is 2925.

Problem: Calculate the value of the expression

35 x 125

Method: We can use the Tails of 5 method to help solve this problem.

The average of the two head will either be a whole number or contain the fraction ½. If the average of the two heads contains the fraction ½, then instead of ending in 25, the product will end in 75 (as 100 times ½ is 50).

To multiply 35 times 125, we observe that both numbers end in 5 which means that we can use the Tails of 5 method to calculate the product. We multiply the heads of the two numbers, 3 and 12, to get 36. Next we take the average of 3 and 12, and we get 7½. The ½ means that the product will end in 75 rather than 25. We add the 7 from the average of the two heads to the product of the two heads, 36, and we get 43.

So, the product of 35 times 125 is equal to 4375.