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

Given:

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

 To do:

We have to find the numbers a and b.

Solution:

According to the question,

$a+b=15$

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

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

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

$\frac{a+b}{ab}=\frac{3}{10}$

$10(15)=3(ab)$

$150=3ab$

$3(15-b)b-150=0$    (Since $a=15-b$)

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

$3(15b-b^2-50)=0$

$15b-b^2-50=0$

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

Solving for $b$ by factorization method,

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

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

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

$b-10=0$ or $b-5=0$

$b=10$ or $b=5$

If $b=10$, then $a=15-10=5$

If $b=5$, then $a=15-5=10$

The numbers a and b are $5$ and $10$ or $10$ and $5$.

Tutorialspoint

e

Updated on: 10-Oct-2022

31 Views

Kickstart Your Career

Get certified by completing the course

Get Started