Question:
The sum of a number and its reciprocal is $2 \frac{1}{20}$. The number is
(a) $\frac{5}{4}$ or $\frac{4}{5}$
(b) $\frac{4}{3}$ or $\frac{3}{4}$
(c) $\frac{5}{6}$ or $\frac{6}{5}$
(d) $\frac{1}{6}$ or 6
Solution:
(a) $\frac{5}{4}$ or $\frac{4}{5}$
Let the required number be $x$.
According to the question:
$x+\frac{1}{x}=\frac{41}{20}$
$\Rightarrow \frac{x^{2}+1}{x}=\frac{41}{20}$
$\Rightarrow 20 x^{2}-41 x+20=0$
$\Rightarrow 20 x^{2}-25 x-16 x+20=0$
$\Rightarrow 5 x(4 x-5)-4(4 x-5)=0$
$\Rightarrow(4 x-5)(5 x-4)=0$
$\Rightarrow x=\frac{5}{4}$ or $x=\frac{4}{5}$
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