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.

1996 AJHSME (Problems • Answer Key • Resources)
Preceded by Problem 10	Followed by Problem 12
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
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1996 AJHSME Problems/Problem 11

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