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Vedic Sutra - All from 9 and Last from 10

Age 14 to 16 Challenge Level:

With this method you only ever need multiplication tables up to 5 times 5. It is one of many ancient Indian sutras and this one involves a cross subtraction method which, according to old historical traditions, is responsible for the acceptance of the ´ mark as the sign of multiplication. Here is a very simple example of the method. Can you give a good explanation of WHY it works?

Suppose we want to multiply 9 by 7. We subtract each number from 10 and, using these differences (or deficiencies), write:

9-1
7-3
6 3

The product has two parts, left and right.
To get the right part (or units digit) multiply the deficiencies (1×3)
The left hand digit (tens digit) of the answer can be found in four different ways. Why do they all give the same answer?

  1. Subtract 10 from the sum of the two given numbers (9+7=16, 16-10=6)
  2. Subtract the sum of the two deficiencies (1+3=4) from 10 and you get 6.
  3. Cross subtract (9-3=6)
  4. Cross subtract (7-1=6)

This gives the answer 63.

Here are some more examples. Try some of your own.

9-1 8-2 9-1 8-2
6-4 7-3 9-1 5-5
5 4 5 6 8 1 4 0
Note: Here you
have to express
5 times 2 as
1 ten and 0 units.