Question

If sum of the two numbers is $18$ the sum of their reciprocals is $\frac{1}{4}$, then find the numbers.

Answer 1

Let two numbers be $a$ and $b$ such that $a+b=18$

Given that,

$\frac{1}{a}+\frac{1}{b}=\frac{1}{4}$

$\Rightarrow \frac{a+b}{ab}=\frac{1}{4}$

Substituting $a+b=18$, we get

$\therefore ab=72$

$a+b=18$

Squaring both sides, we get

$(a+b{)}^2=324$

$\Rightarrow a^2+b^2+2ab=324$

Substituting value of $ab$, we get

$\Rightarrow a^2+b^2=324-2\left(72\right)=180$

$\therefore a^2+b^2-2ab=(a-b{)}^2=180-144=36$

$\therefore (a-b)=6$

$\Rightarrow a+b=18$ and $a-b=6$

Solving these equations, we get

$a=12,b=6$