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

To solve this problem, we are given three boxes containing all the rare jewels owned by the king, distributed in a specific way:

1. The first box contains \(\frac{1}{3}\) of the total jewels.
2. The second box contains \(\frac{k}{5}\) of the total jewels, where \(k\) is some positive integer.
3. The third box contains 66 jewels.

Let's denote the total number of rare jewels as \(x\).

According to the information we have:

• The jewels in the first box: \(\frac{x}{3}\)
• The jewels in the second box: \(\frac{kx}{5}\)

The sum of jewels in all three boxes must equal the total jewels:

x = \(\frac{x}{3} + \frac{kx}{5} + 66\)

To solve for \(x\), we first eliminate the fractions by multiplying through by 15 (the LCM of 3 and 5):

15x = 5x + 3kx + 990

Rearrange to collect \(x\) terms on one side:

15x - 5x - 3kx = 990

This simplifies to:

x(10 - 3k) = 990

Solving for \(x\) gives:

x = \(\frac{990}{10 - 3k}\)

We are told that \(x\) must be a positive integer, so \(990\) must be divisible by \((10 - 3k)\).

Checking for feasible values of \(k\) that make this expression valid:

• When \(k = 2\), \(10 - 3k = 4\): \(x = \frac{990}{4}\) is not an integer.
• When \(k = 3\), \(10 - 3k = 1\): \(x = \frac{990}{1} = 990\), which is an integer.

Thus, with \(k = 3\), the total number of rare jewels \(x\) is 990. This satisfies the integer condition and explains how the distribution maintains 66 jewels in the third box.

Therefore, the correct answer is:

990

Answer 2

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