1985 AJHSME Problem 22

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Problem

Assume every 7-digit whole number is a possible telephone number except those that begin with $0$ or $1$. What fraction of telephone numbers begin with $9$ and end with $0$?

$\text{(A)}\ \frac{1}{63} \qquad \text{(B)}\ \frac{1}{80} \qquad \text{(C)}\ \frac{1}{81} \qquad \text{(D)}\ \frac{1}{90} \qquad \text{(E)}\ \frac{1}{100}$

Note: All telephone numbers are 7-digit whole numbers.

Solution

There are a total of $8 \cdot 10^6$ possible phone numbers. The first number is 9 and the last number is 0, so two numbers are fixed, making it so that $10^5$ numbers satisfy the conditions. So, the answer is $\frac{10^5}{8 \cdot 10^6} = \boxed{\frac{1}{80}}.$ The answer is B.

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