# 2016 June

Convert to improper fractions:

65/7  x 68/7 =

Perform the multiplication: (May require pen and paper or calculator)

65/7 x 68/7 = 4420 /49

Convert result to mixed fraction: (May require pen and paper or calculator)

4420 /49 =  90 10/49

MATH-Inic Mental Solution:

The factors 9 2/7 and 9 5/7 are “complementary” in the sense that they have the same whole number part and their fractional parts 2/7 and 5/7 total to 1.

Apply the Vedic Math Sutra “By one more than the one before“:

a)Multiply the whole number part by the next higher number to get the whole number part of the answer

b) Multply the fractional parts to get the fractional part of the answer.

9 x (9 + 1) | 2/7 x 5/7 = 90 | 10/49 = 90 10/49

The solution can be better understood if we use the distributive laws:

9 2/7  x 9 5/7 = (9 + 2/7) x ( 9 5/7)

= 9(9) + 9(5/7) + 2/7(9) + 2/7(5/7)

= 9(9)  + 9(5/7+ 2/7) + (2/7)(5/7)

= 9(9) + 9(1) + 10/49

= 9(9 +1) + 10/49

90 10/49

This solution is similar to that of our previous post “8 1/2 squared“,which is shown below.

For more examples and exercises, please see MSC # 16 -Multiplying Complementary Numbers, pp 57-59, 25 Math Short Cuts

8 1/ 2 Squared

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## What is 8 1/2 x 8 1/2 ?

Convert to improper fractions:

8 ½ x 8 ½  = 17/2 x 17/2

Perform the multiplication:

17/2 x 17/2  = (17 x 17) /(2 x 2)

= 289/ 4

Convert to Mixed number: 72  1/ 4

### MATH-Inic Mental solution:

This is an extension of the technique of squaring numbers ending in 5 where the answer is composed of two parts. The first part is obtained by using the Vedic Math Sutra “ By one more than the one before” and the second part is always 25.  For example 25 x 25 is 2 x (2+1)| 5 x 5 = 625.

Here the answer is simply 8 x (8 + 1)|1/2 x 1/ 2 = 72|1/4 = 72 1/4

### Algebraic proof:

Use (a + b)2 = a2 + 2(a)(b) + b2

(a + 1/2)2 = a2 + 2(a)(1/2) + (1/2)2

= a2 + a + (½)2

= a(a+1)+ ¼

For further explanations and more examples see MSC #15 Squaring numbers ending in 5, p 53, 25 Math Short Cuts

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