Find the average of all multiples of 7 between 7 and 777, inclusive. Answer: 392
This is like the Gaussian problems (adding consecutive numbers) as it is the pairs of numbers that allow us to more easily solve these types of problems.
For example, adding the numbers 1 - 10 can be done by grouping the smallest and largest 1 and 10 to make 11, (then the next smallest and largest) 2 and 9 to make 11, 3 and 8, 4 and 7 and 5 and 6 to get 5 pairs of 11. So the sum of the numbers form 1 to 10 is 5 x 11 = 55.
From 7 to 777 inclusive is just like 1 to 111 inclusive (divide each by 7).
One of Math educator and writer Polya's methods: Solve a Simpler Problem. Step 2: Devise a Plan
Choosing easier numbers can often make the solution easier and simpler to understand.
To find the average of 1 to 111, find the average of each pair -- the average of 1 and 111 is 56 (1+111)/2.
The average of 2 and 110 is 56 and so on.
So the 392 is the average of 7 and 777 and also the average of 14 and 770 (the next two multiples) is also 392.
Please visit the Excel spreadsheet on docstoc and scroll down to see that the answer is 392!!
http://www.docstoc.com/docs/45498081/Math-Confidence-Brain-Teaser-June-2010
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