# 1996 AJHSME Problems/Problem 11

## Problem

Let $x$ be the number $$0.\underbrace{0000...0000}_{1996\text{ zeros}}1,$$ where there are 1996 zeros after the decimal point. Which of the following expressions represents the largest number? $\text{(A)}\ 3+x \qquad \text{(B)}\ 3-x \qquad \text{(C)}\ 3\cdot x \qquad \text{(D)}\ 3/x \qquad \text{(E)}\ x/3$

## Solution

Estimate each of the options. $A$ will give a number that is just over $3$. $B$ will give a number that is just under $3$. This eliminates $B$, because $A$ is bigger. $C$ will give a number that is barely over $0$, since it is three times a tiny number. This eliminates $C$, because $A$ is bigger. $D$ will give a huge number. $\frac{1}{x}$ will get very, very large in magnitude when $x$ gets close to zero. You can see this by examining the sequence $x=0.1$, which gives $10$ as the reciprocal, $x=0.01$, which gives $100$ as the reciprocal, and $x=0.001$, which gives $1000$ as the reciprocal. Thus, $D$ will be huge, and this eliminates $A$. $E$ will give a small number, since you're dividing a tiny number into thirds. This eliminates $E$, and gives $\boxed{D}$ as the answer.

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