# Problem of the Day

The ratio of cats to dogs at a pet store is 11:15. Â The ratio of dogs to hamsters is 3:11. Â Let be the total number of cats, dogs, and hamsters. What is ?

(1) (2) is an integer.

C. Both statements together are sufficient, but neither statement alone is sufficient.

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### Solution

[latexpage]

First, make a three-way ratio that expresses the number of cats:dogs:hamsters. Â To do this, multiply the ratio of dogs to hamsters, 3:11, by 5 to get 15:55. Â Note that this ratio is equivalent to the original ratio, but now that the number of dogs is the same in both ratios, we can merge the two ratios into a three-way ratio. Â If the ratio of cats to dogs is 11:15 and the ratio of dogs to hamsters is 15:55, then the ratio of cats to dogs to hamsters is 11:15:55. Â The sum of these three numbers is 81. Â This tells us that the total number of Â cats, dogs, and hamsters must be a multiple of 81. Â In other words, $T$ is a multiple of 81.

Statement #1: This tells us that $T<300$. Â We also know that $T$ is a multiple of 81. Â This means that $T$ could be 81, 162, or 243. Â Obviously these three numbers have different square roots, so statement 1 is insufficient by itself.

Statement #2: This tells us that $\sqrt{T}$ is an integer. Â In other words, $T$ is a perfect square. Â However, infinitely many multiples of 81 are perfect squares. Â For example $\sqrt{81}=9$ and $sqrt{324}=18$. Â Thus, statement 2 is insufficent on its own.

However, both statements together ARE sufficient to answer the question because 81 is the only multiple of 81 less then 300 that is a perfect square. Â To see that this is true, consider the following:

$\sqrt{81}=9$
$\sqrt{81 \cdot 2}=9\sqrt{2}$
$\sqrt{81 \cdot 3}=9\sqrt{3}$
$\sqrt{81 \cdot 4}=9\sqrt{4}=9 \cdot 2=18$

But $81 \cdot 4=324$, so that violates statement 1. Â Hence, 81 is the only multiple of 81 less than 300 that is a perfect square.