2006 AIME A Problems/Problem 7

Problem

An angle is drawn on a set of equally spaced parallel lines as shown. The ratio of the area of shaded region $\mathcal{C}$ to the area of shaded region $\mathcal{B}$ is 11/5. Find the ratio of shaded region $\mathcal{D}$ to the area of shaded region $\mathcal{A}.$

Solution

There are $\frac{1000}{10} = 100$ numbers up to 1000 that have 0 as their units digit. All of the other excluded possibilities for $a$ are when $a$ or $b$ have a 0 in the tens digit, or if the tens digit of $a$ is 0 or 9. Excluding the hundreds, which were counted above already, there are $9 \cdot 2 = 18$ numbers in every hundred numbers that have a tens digit of 0 or 9, totaling $10 \cdot 18 = 180$ such numbers. However, the numbers from 1 to 9 and 991 to 999 do not have 0s, so we must subtract $18$ from that to get $162$ . Therefore, there are $1000 - (100 + 162) = 738$ such ordered pairs.

2006 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 6	Followed by Problem 8
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2006 AIME A Problems/Problem 7

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