Question

If we divide $40$ into two parts such that the sum of their reciprocals is $\frac{8}{75}$, then numbers are $15$ and $25$.

• A. Yes
• B. No
• C. Ambiguous
• D. Data insufficient

Answer 1

The correct option is A Yes

Let the numbers be $a$ and $b$
$a+b=40$

$\therefore a=40-b$
Sum of their reciprocals is $\frac{8}{75}$

$\frac{1}{a}+\frac{1}{b}=\frac{8}{75}$

$\frac{a+b}{ab}=\frac{8}{75}$

$ab=\frac{75\times 40}{8}=375$

$b(40-b)=375$

$b^2-40b+375=0$

$b^2-25b-15b+375=0$

$b(b-25)-15(b-25)=0$

$(b-15)(b-25)=0$

$b=15,25$
Hence, the two numbers are $15$ and $25$.