Rapidly square any number ending in 5

Book cover for Rapid Math Tricks and TipsA couple of days ago I blogged about this book: Rapid Math Tricks and Tips: 30 Days to Number Power by Edward H. Julius (Wiley, 1992) ISBN 0-471-57563-1. I’ve had quite a few people saying how fun that last tip was (multiplying by 99). Here’s my favourite trick of all in the first book: how to square any number that ends in 5. (Remember that the symbol (^) means raise to the power, so 5^2 is 5 raised to the power of 2, or 5 x 5.)

Here’s what Edward H. Julius has to say:

Strategy: This trick is one of the oldest in the book, and one of the best! To square a number that ends in 5,

  1. first multiply the tens digit by the next whole number.
  2. To that product, affix the number 25.

The number to affix (25) is easy to remember, because 5^2 = 25. Although a calculation such as 7.5 x 750 is technically not a square, it too can be solved using this technique. This trick will also work for numbers with more than two digits.

For example: 15^2:

  1. Multiply the tens digit (1) by the next whole number (2): 1 x 2 = 2.
  2. Affix 25 to that answer: 225 (the answer).

Another example, 65^2:

  1. Multiply 6 x 7 = 42
  2. Affix 25: 4,225 (the answer)

How about, 450^2:

  1. Ignore the zero and think 45^2
  2. Multiply 4 x 5 = 20
  3. Affix 25: 2025
  4. For each zero initially disregarded in a squaring problem two must eventually be affixed to obtain the product.
  5. Affird two zeros to the end: 202,500.

And what about the not-squared problem? This method only works if the two numbers have the same digits, eg 65 x 6.5 or 950 x 9.5 but not 65 x 3.5. So for 65 x 6.5:

  1. Disregard the decimal point for now
  2. Multiply 6 x 7 = 42
  3. Affix 25: 4225
  4. Applying a test of reasonableness you can see that 65 x 5 would be 325, so x 6.5 must be between 325 and 650 (which is 65 x 10). So re-inserting the decimal point gives the answer: 422.5.

How cool is that! Now, go buy the book! It’s only £6.96 at Amazon UK.

Published by

Gareth Saunders

I’m Gareth J M Saunders, 46 years old, 6′ 4″, father of 3 boys (including twins). Latterly, web architect and agile project manager at the University of St Andrews and warden at Agnes Blackadder Hall. Currently on sabbatical. I am a priest in the Scottish Episcopal Church, and I sing with the NYCGB alumni choir.

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