Question

Sum of two numbers is $15$, if sum of their reciprocals is $\frac{3}{10}$. Find the numbers.

Answer 1

Let the numbers are $xandy$

According to given question,

$x+y=15$ ……. (1)

Again, According to given question,

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

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

$\frac{15}{xy}=\frac{3}{10}$

$xy=50$

$y=\frac{50}{x}$ …… (2)

By equation (1) and (2) to, we get

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

$x^2+50=15x$

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

$x^2-(10+5)x+50=0$

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

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

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

Then, $x=5,10$

By equation (1) to,

put $x=5$ and we get,

$x+y=15$

$5+y=15$

$y=15$

Put $x=10$ and we get,

$x+y=15$

$10+y=15$

$y=15-10=5$

Then,$(5,10)or(10,5)$ is the answer.