Question

A natural number, when increased by $12$, equals $160$ times its reciprocal. Find the number.

Answer 1

Step 1: Let the number is $n$and form the equation.

Know that, when the number is increased by $12$, equals $160$ times its reciprocal.

Therefore,

$\begin{array}{l}n+12=\frac{1}{n}\times 160\\ \Rightarrow n^2+12n=160\\ \Rightarrow n^2+12n-160=0\end{array}$

Step 2: Solve the above quadratic equation.

$\begin{array}{l}{n}^2+12n-160=0\\ \Rightarrow n^2+20n-8n-160=0\\ \Rightarrow n\left(n+20\right)-8\left(n+20\right)=0\\ \Rightarrow \left(n+20\right)\left(n-8\right)=0\\ \Rightarrow n=-20,8\end{array}$

Since natural numbers cannot be negative.

Therefore $n=8$.

Hence the natural number when increased by $12$, equals $160$ times its reciprocal is $8$.