Question

The difference of two natural numbers is $4$ and the difference of their reciprocals is $\frac{1}{3}$. Find the numbers.

• A. $y=4$ $x=8$
• B. $y=2$ $x=6$
• C. $y=10$ $x=6$
• D. $y=8$ $x=12$

Answer 1

The correct option is B $y=2$ $x=6$

$letnumberbexandythen|x-y|=4orx-y=4(letx>y)and\left|\right(\frac{1}{x})-(\frac{1}{y}\left)\right|=\left(\frac{1}{3}\right)or\left(\frac{1}{y}\right)-\left(\frac{1}{x}\right)=\left(\frac{1}{3}\right)\left(as\right(\frac{1}{y})>(\frac{1}{x}\left)\right)or\left(\frac{x-y}{xy}\right)=\left(\frac{1}{3}\right)\left(\frac{4}{xy}\right)=\left(\frac{1}{3}\right)\therefore xy=12theny=\left(\frac{12}{x}\right)nowx-\left(\frac{12}{x}\right)=4\left(\frac{x^2-12}{x}\right)=4x^2-12=4x{x}^2-4x-12=0(x-6)(x+2)=0\therefore x=6theny=\left(\frac{12}{6}\right)=2$