Question

The sum of two natural numbers is 8 and the difference of their reciprocals is $2/15$. Find the numbers.

Answer 1

Let us consider two numbers 'x' and 'y'.
So according to the question,
$\frac{1}{x}-\frac{1}{y}=\frac{2}{15}\dots ..\left(i\right)$
It is given that, $x+y=8$
(ii) $So,y=8-x\dots$
Now, substitute the value of y in equation (i),
we get $\frac{1}{x}-\frac{1}{8-x}=\frac{2}{15}$
By taking LCM,

$\frac{8-x-x}{x(8-x)}=\frac{2}{15}$

$\frac{(8-2x}{x(8-x)}=\frac{2}{15}$

By cross multiplying,
$15(8-2x)=2x(8-x)$
$120-30x=16x-2x^2$
$120-30x-16x+2x^2=0$
$2x^2-46x+120=0$
Divide by $2,$
we get $x^2-23x+60=0$
let us factorize,
$x^2-20x-3x+60=0$
$x(x-20)-3(x-20)=0$
$(x-20)(x-3)=0$
$So$
$(x-20)=0$ or $(x-3)=0$
$x=20$ or $x=3$
$Now$
Sum of two natural numbers, $y=8-x=8-20=-12,$
which is a negative value.
So value of $x=3,y=8-x=8-3=5$
$\therefore$ The value of $x$ and $y$ are 3 and 5