# 1997 AJHSME Problems/Problem 16

## Problem

Penni Precisely buys \$100 worth of stock in each of three companies: Alabama Almonds, Boston Beans, and California Cauliflower. After one year, AA was up 20%, BB was down 25%, and CC was unchanged. For the second year, AA was down 20% from the previous year, BB was up 25% from the previous year, and CC was unchanged. If A, B, and C are the final values of the stock, then $\text{(A)}\ A=B=C \qquad \text{(B)}\ A=B $\text{(D)}\ A

## Solution 1

AA is $100 \cdot 120\% = 100\cdot 1.2 = 120$ after one year. After the second year, AA is $120\cdot 0.8 = 96$.

BB is $100 \cdot 75\% = 100\cdot 0.75 = 75$ after one year. After the second year, BB is $75\cdot 1.25 = 93.75$

CC remains unchanged throughout, and stays at $100$.

Thus, $B < A < C$, and the right answer is $\boxed{E}$

## Solution 2

AA will be $100 \cdot \frac{6}{5} \cdot \frac{4}{5} = 100\cdot \frac{24}{25}$ at the end.

BB will be $100 \cdot \frac{3}{4} \cdot \frac{5}{4} = 100 \cdot \frac{15}{16}$ at the end.

CC will be unchanged at $100$.

Since all the fractions are under $1$, $C$ will be highest value.

Since $\frac{24}{25}$ is only $\frac{1}{25}$ away from $1$, while $\frac{15}{16}$ is $\frac{1}{16}$ away from $1$, $\frac{24}{25}$ is closer to $1$, and will be closer to the original value.

Thus, $B < A < C$, and the right answer is $\boxed{E}$

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