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Problem 794

AMC 12 late, AIME early

Algebra Difficulty 4.3 Multiple choice Slovenia — National Math Olympiad · Slovenia · 2012

Peter breeds horses and cows on his farm. One day, the number of horses was the same as the number of cows and greater than $0$ . Then Peter bought some more cows so that the number of cows increased by $50\%$ . Now only $30\%$ of the animals are horses. How many horses does Peter have on his farm?

Pick one

Official solution

Say at the beginning there were $x$ cows and $x$ horses on the farm. When the number of cows increased by $50\%$ , there were $x + \frac{1}{2}x = \frac{3}{2}x$ cows. Hence the ratio of horses was equal to $x/(x + \frac{3}{2}x) = \frac{2}{5}$ , which is $40\%$ and not $30\%$ . The correct answer is thus (E).

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