Adding a Long List of Numbers

For some people, adding a list of not very different numbers is a frequent task. To me this happens every time I fill my tax returns; there is always a few dozen professional books I buy over a year that I consider deductible. Of course, it is possible to use a spreadsheet or a specialized application, but I am accustomed to doing this the old fashioned way. Using a calculator is far from being fool proof; an innocent typo may flash a red light during an audit.

Teachers who have to compute test grade averages might also find the technique useful.

So assume we need to compute the sum of 16 numbers:

97 + 86 + 83 + 95 + 85 + 70 + 84 + 72 + 77 + 81 + 70 + 85 + 84 + 76 + 92 + 66.

Estimate a possible average of the numbers in the sum. For the given example, I'd choose 80. Instead of adding the given numbers, we shall add the differences of these numbers and the chosen average estimate:

17 + 6 + 3 + 15 + 5 + (-10) + 4 + (-8) + (-3) + 1 + (-10) + 5 + 4 + (-4) + 12 + (-14).

At a glance, you can see that, say, 15 + 5 cancels (-10) + (-10), and 4 cancels (-4), leaving a shorter sum

17 + 6 + 3 + 4 + (-8) + (-3) + 1 + 5 + 12 + (-14) = 30 + (-11) + 18 + (-14) = 19 + 4 = 23.

The "dropped" part of the original sum is 80·16 = 800 + 480 = 1280, making the total 1280 + 23 = 1303.

The trick here is to avoid dealing with large accumulations and a need to memorize intermediate results. Always scan the sums for possible cancellations. For example, in the last sum we may have noticed that 3 and (-3) cancel out and so do 3 + 5 and (-8). Taking into account the latter we would get a shorter sum

17 + 6 + 4 + (-3) + 1 + 12 + (-14),

in which it is hard to fail to notice that 17 and (-3) + (-14) also cancel out

17 + 6 + 4 + (-3) + 1 + 12 + (-14) = 6 + 4 + 1 + 12 = 10 + 13 = 23.

With a little practice, such shortcuts pop into view automatically.

References

  1. Shai Simonson, Rediscovering Mathematics: You Do the Math, MAA, 2011, p. 13

Related material
Read more...

  • Multiplication by 9, 99, 999, (Multiply + Subtract) etc.
  • Squaring 2-Digit Numbers
  • Division by 5
  • Multiplication by 2
  • Multiplication by 5
  • Multiplication by 9, 99, 999, etc. (Something Special)
  • Product of 10a + b and 10a + c where b + c = 10
  • Product of numbers close to 100
  • Product of two one-digit numbers greater than 5
  • Product of 2-digit numbers
  • Squaring Numbers in Range 26-50
  • Squaring Numbers in Range 51-100
  • Squares of Numbers That End in 5
  • Squares Can Be Computed Squentially
  • How to Compute Fast Any Square

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