Question

The sum of two numbers is $18$. The sum of their reciprocal is $\frac{1}{4}$. Find the numbers.

• A. The numbers are $18$ and $4$
• B. The numbers are $12$ and $6$
• C. The numbers are $17$ and $14$
• D. The numbers are $10$ and $14$

Answer 1

The correct option is B The numbers are $12$ and $6$

Let the two numbers be $x$ and $y$

Sum of two numbers are:
$x+y=18$ ........ (1)

Also,
$\frac{1}{x}+\frac{1}{y}=\frac{1}{4}$.......(2)

$\frac{y+x}{xy}=\frac{1}{4}$

Put $x+y=18$

$\frac{18}{xy}=\frac{1}{4}\Rightarrow xy=72$

We know that ${(x-y)}^2={(x+y)}^{2}4xy$

${(x-y)}^2={\left(18\right)}^2-4\times 72$

${(x-y)}^2=36xy=6$ ....... (3)

Adding equation (1) and (3)
$2x=24$
$x=12$
Substitute $x=12$ in equation(1)
$12+y=18$
$y=6$
Hence, the numbers are $12$ and $6$