To solve this problem, we need to determine the ratio of the duration of A, B, and C's investments given that their profit sharing ratio is the same. The relationship between their investment amounts, durations, and the profits earned forms the basis of our solution.

Given:

• The ratio of investments by A, B, and C is \(2 : 3 : 5\).
• The profits earned by A, B, and C are the same.
• Profit is proportional to the product of the amount and the duration of the investment.

Since the profits are the same, the product of investment and duration for A, B, and C must be equal. Let's denote the duration of A, B, and C's investments as \(x\), \(y\), and \(z\) respectively.

From the profit condition:

\(2x = 3y = 5z\)

Now, express each equality in terms of a constant \(k\):

• \(2x = k\) implies \(x = \frac{k}{2}\)
• \(3y = k\) implies \(y = \frac{k}{3}\)
• \(5z = k\) implies \(z = \frac{k}{5}\)

Hence, the ratio of the durations of their investments is:

\(x : y : z = \frac{k}{2} : \frac{k}{3} : \frac{k}{5}\)

To simplify, find a common denominator for the fractions, which is 30 in this case:

• \(\frac{k}{2} = \frac{15k}{30}\)
• \(\frac{k}{3} = \frac{10k}{30}\)
• \(\frac{k}{5} = \frac{6k}{30}\)

Therefore, the ratio of the durations of their investments is:

\(15 : 10 : 6\)

The correct answer is thus, the ratio of the duration of their investments is 15 : 10 : 6.