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# Vedic Sutra - All from 9 and Last from 10

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Age 14 to 16

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- Problem
- Student Solutions

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?

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