Problem of the Day

Solution

[latexpage]

First, make a three-way ratio that expresses the number of cats:dogs:hamsters.  To do this, multiply the ratio of dogs to hamsters, 3:11, by 5 to get 15:55.  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.  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.  The sum of these three numbers is 81.  This tells us that the total number of  cats, dogs, and hamsters must be a multiple of 81.  In other words, $T$ is a multiple of 81.

Statement #1: This tells us that $T<300$.  We also know that $T$ is a multiple of 81.  This means that $T$ could be 81, 162, or 243.  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.  In other words, $T$ is a perfect square.  However, infinitely many multiples of 81 are perfect squares.  For example $\sqrt{81}=9$ and $sqrt{324}=18$.  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.  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.  Hence, 81 is the only multiple of 81 less than 300 that is a perfect square.