Question

the sum of 2 natural no. is 8 determine the no. if the sum of their reciprocal is 8/15

Answer 1

Let,

One number = x

Another number = 8 - x

A.T.Q.

$\sf{{\dfrac{1}{x}} + {\dfrac{1}{8 - x}} = {\dfrac{8}{15}}}$

$\sf{{\dfrac{8 - x + x}{(x)(8 - x)}} = {\dfrac{8}{15}}}$

$\sf{{\dfrac{8}{(x)(8 - x)}} = {\dfrac{8}{15}}}$

$\sf{{\dfrac{{\cancel{8}}}{(x)(8 - x)}} = {\dfrac{{\cancel{8}}}{15}}}$

$\sf{{\dfrac{1}{(x)(8 - x)}} = {\dfrac{1}{15}}}$

Cross Multiplication

15 = (x)(8 - x)

15 = 8x - x²

x² - 8x + 15 = 0

By Middle Term Factorisation

x² - 3x - 5x + 15 = 0

Taking common

x ( x - 3 ) - 5 ( x - 3 ) = 0

( x - 5 ) ( x - 3 ) = 0

Using Zero Product Rule

x - 5 = 0 and x - 3 = 0

x = 5 and x = 3

$<u>Case I : When x = 5</u>$

One number = x = 5

Another number = 8 - x = 8 - 5 = 3

$<u>Case II : When x = 3</u>$

One number = x = 3

Another number = 8 - x = 8 - 3 = 5

Hence,

$<b>The numbers are 3 and 5.</b>$