Question

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

• A. $a=5,b=10$
• B. $a=10,b=5$
• C. $a=10,b=-5$
• D. both a and b

Answer 1

Answer: (D)

both a and b

$\Rightarrow$ The two numbers are $a$ and $b$

According to the question,

$\Rightarrow$ $a+b=15$

$\Rightarrow$ $a=15-b$ ---- ( 1 )

$\Rightarrow$ $\frac{1}{a}+\frac{1}{b}=\frac{3}{10}$

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

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

$\Rightarrow$ $150=3b(15-b)$

$\Rightarrow$ $50=b(15-b)$

$\Rightarrow$ $50=15b-b^2$

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

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

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

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

$\therefore$ $b=5$ or $b=10$

Putting both values of $b$ in equation ( 1),

$\Rightarrow$ $a=15-5$ or $a=15-10$

$\therefore$ $a=10$ or $a=5$