# 1989 AJHSME Problems/Problem 18

## Problem

Many calculators have a reciprocal key $\boxed{\frac{1}{x}}$ that replaces the current number displayed with its reciprocal. For example, if the display is $\boxed{00004}$ and the $\boxed{\frac{1}{x}}$ key is depressed, then the display becomes $\boxed{000.25}$. If $\boxed{00032}$ is currently displayed, what is the fewest number of times you must depress the $\boxed{\frac{1}{x}}$ key so the display again reads $\boxed{00032}$? $\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ 5$

## Solution

Let $f(x)=\frac{1}{x}$. We have $$f(f(x))=\frac{1}{\frac{1}{x}}=x$$ Thus, we need to iterate the key pressing twice to get the display back to the original $\rightarrow \boxed{\text{B}}$.

## Solution 2 (simpler, no use of functions)

In terms of mathematics, a number will always be changed back to its original number if you flip, or "reciprocal" it, twice. Therefore, no matter what number it will display, the answer will always be $2 = \boxed{\text{B}}.$

~DuoDuoling0