Question

The sum of two numbers is 8 determine the numbers if the sum of their reciprocal is 8 by 15

Answer 1

Let the two natural numbers be x and y

X+y=8

X=8-y ---(Equation-1)

$\frac{1}{x}+\frac{1}{y}=\frac{8}{15}$ ---(Equation -2)

We get,

$\frac{1}{x}+\frac{1}{y}=\frac{8}{15}$

$\frac{1}{8-y}+\frac{1}{y}=\frac{8}{15}$

$\frac{y+8-y}{(8-y)y}=\frac{8}{15}$

${\frac{8}{{8y-y^2}^{}}}^{}=\frac{8}{15}$

-8y²+64y=120

-8y²+64y-120=0

8y²-64y+120=0

y²-8y+15=0

y²-5y-3y+15=0

(sum=-8,Product=15)

y(y-5)-3(y-5)=0

y-5=0 (or)y-3=0

y=5 (or) y=3

If y=5 then x=8-5

X=3

If y=3 then x=8-3

X=5

Therefore the two natural numbers are 3and 5