Question

Divide $15$ into two parts such that the sum of their reciprocals is $\frac{3}{10}$.

Answer 1

Suppose that,

The two parts is $x$ and $y$.

So,

According to question,

$x+y=15$ …… (1)

Then,

$\frac{1}{x}+\frac{1}{y}=\frac{3}{10}$

$\Rightarrow \frac{x+y}{xy}=\frac{3}{10}$

$\Rightarrow 10x+10y=3xy$

$\Rightarrow 10(x+y)=3xy$

$\Rightarrow 10\times 15=3xy$

$\Rightarrow xy=10\times 5$

$\Rightarrow xy=50$

$\Rightarrow y=\frac{50}{x}......\left(2\right)$

Put in (1) and we get,

$x+\frac{50}{x}=15$

$\Rightarrow x^2+50=15x$

$\Rightarrow x^2-15x+50=0$

$\Rightarrow x^2-10x-5x+50=0$

$\Rightarrow x(x-10)-5(x-10)=0$

$\Rightarrow (x-10)(x-5)=0$

$\Rightarrow x=5,10$

If put $x=5,soy=10$

If put$x=10so,y=5$

Hence, this is the answer.