Question

The sum of two numbers a and $b$ is $15$ and the sum of their reciprocal $\frac{1}{a}$ and $\frac{1}{b}$ is $\frac{3}{10}$. Find $a$ and $b$.

Answer 1

$a+b=15$

$a=15-b$ (1)

$\frac{1}{a}+\frac{1}{b}=\frac{3}{10}$ (2)

Substituting (1) in (2)

$\frac{1}{15-b}+\frac{1}{b}=\frac{3}{10}$

$\frac{b+15-b}{\left(b\right)(15-b)}=\frac{3}{10}$

$\frac{15}{15b-b^2}=\frac{3}{10}$

$150=45b-3b^2$

$3b^2-45b+150=0$

$b=\frac{45\pm \sqrt{(45{)}^2-4\times 3\times 150}}{2\times 3}$

$b=\frac{45\pm \sqrt{2025-1800}}{2\times 3}$

$b=\frac{45\pm \sqrt{225}}{2\times 3}$

$b=\frac{45\pm 15}{2\times 3}$

$b=\frac{60}{6}$ or $b=\frac{30}{6}$

$b=10$ or $b=5$

Hence the numbers are 10 and 5.