Question

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

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Solution

## 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,$\begin{array}{l}n+12=\frac{1}{n}×160\\ ⇒{n}^{2}+12n=160\\ ⇒{n}^{2}+12n-160=0\end{array}$Step 2: Solve the above quadratic equation.$\begin{array}{l}{n}^{2}+12n-160=0\\ ⇒{n}^{2}+20n-8n-160=0\\ ⇒n\left(n+20\right)-8\left(n+20\right)=0\\ ⇒\left(n+20\right)\left(n-8\right)=0\\ ⇒n=-20,8\end{array}$Since natural numbers cannot be negative.Therefore $n=8$.Hence the natural number when increased by $12$, equals $160$ times its reciprocal is $8$.

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