### Mental Math - Summing by Pairs

Problem: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = _____ .

Method: Using pairs of number, we combine the first with the last, the second with the next to last, and so on. Each pair adds up to 11 and we have five pairs of numbers. This means that the total is 5 x 11 = 55. This makes the problem much easier especially when you have many numbers in the series.

Let’s take a look, 1 + 10 = 11 is the sum of one pair,
2 + 9 = 11 is the sum of the second pair,
3 + 8 = 11 is the sum of the third pair,
4 + 7 = 11 is the sum of the fourth pair,
5 + 6 = 11 is the sum of the fifth pair.

Then we can either add 11 + 11 + 11 + 11 + 11 = 55 or multiply 5 x 11 = 55.

Problem: 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = _____ .

Method: Using pairs of numbers, we group the first and last numbers, the second and next to last number, and so on. In this case each sum is 20 and there there are five pairs of number giving a total of 100.

In this case we have, 1 + 19 = 20,
3 + 17 = 20,
5 + 15 = 20,
7 + 13 = 20,
9 + 11 = 20

Then we can either add 20 + 20 + 20 + 20 + 20 = 100 or 5 x 20 = 100.