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.

See Also

1996 AJHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 10
Followed by
Problem 12
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All AJHSME/AMC 8 Problems and Solutions

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