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?

Accepted Answer

A. Let $X$ be the total number of jewels. $X/3 + kX/5 + 66 = X \implies 66 = X\left(1 - \frac{1}{3} - \frac{k}{5}ight) = X\left(\frac{2}{3} - \frac{k}{5}ight) = X\frac{10 - 3k}{15}$. So $X(10 - 3k) = 66 	imes 15 = 990$. Since $k$ is a positive integer and $10 - 3k > 0$, possible values for $k$ are $1, 2, 3$. If $k = 1$: $10 - 3 = 7 \implies X = 990/7$ (not an integer). If $k = 2$: $10 - 6 = 4 \implies X = 990/4$ (not an integer). If $k = 3$: $10 - 9 = 1 \implies X = 990$. Thus $X = 990$ is uniquely determined.

King's Rare Jewels in Three Boxes

Explanation

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