Question

A king has distributed all his rare jewels in three boxes. The first box contains 1/3 of the rare jewels, while the second box contains k/5 of the rare jewels, for some positive integer value of k. The third box contains 66 rare jewels.
How many rare jewels does the king have?

• A. 990
• B. 660
• C. 240
• D. 1080
• E. Cannot be determined uniquely from the given information.

Answer 1

The Correct Option is A

Step 1: Represent the total number of jewels. Let the total number of jewels be N. According to the problem:

Jewels in the first box = $\frac{1}{3}$N, Jewels in the second box = $\frac{k}{5}$N, Jewels in the third box = 66.

The total number of jewels is:

N = $\frac{1}{3}$N + $\frac{k}{5}$N + 66.

Step 2: Simplify the equation. Rearrange terms:

N − $\frac{1}{3}$N − $\frac{k}{5}$N = 66.

Combine terms:

$\left( 1 - \frac{1}{3} - \frac{k}{5} \right)$N = 66.

Simplify the coefficients:

$\left( \frac{3}{3} - \frac{1}{3} - \frac{k}{5} \right)$N = 66 = =$>$ $\left( \frac{2}{3} - \frac{k}{5} \right)$N = 66.

Step 3: Solve for k. Since k is a positive integer, test values such that $\frac{2}{3} - \frac{k}{5} $>$ 0$. Let k = 2:

$\frac{2}{3} - \frac{2}{5} = \frac{10}{15} - \frac{6}{15} = \frac{4}{15}$

Substitute into the equation:

$\frac{4}{15}$N = 66 = =$>$ N = $\frac{66 \times 15}{4}$ = 990 jewels.

Answer: 990