Question

The difference of two numbers is $5$ and the difference of their reciprocals is $\frac{1}{10}$. Find the numbers.

• A. $10,50$
• B. $10,5$
• C. 10,−510,−5
• D. $0,5$

Answer 1

The correct option is C 10,−510,−5

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

then, Difference of two numbers is$x-y=5$ .......$\left(1\right)$

Differences of their reciprocals is $\frac{1}{y}-\frac{1}{x}=\frac{1}{10}$

$\Rightarrow \frac{x-y}{xy}=\frac{1}{10}$

$\Rightarrow (x-y)=\frac{xy}{10}$ ....$\left(2\right)$

Substitute the value of $x-y=5$ in eqn$\left(2\right)$ we get

$\Rightarrow 5=\frac{xy}{10}$ or $xy=50$

$\Rightarrow y=\frac{50}{x}$ .....$\left(3\right)$

Substituting the value of $y=\frac{50}{x}$ in eqn$\left(1\right)$ we get

$x-\left(\frac{50}{x}\right)=5$

$\Rightarrow x^2-5x-50=0$

$(x-10)(x+5)=0$

$x=10$ $x=-5$