A and B are partners in a business. A contributes \( \frac{1}{4} \) of the capital for \(15\) months and B receives \( \frac{2}{3} \) of the profit. Find out how long B's money was used. A এবং B একটি ব্যবসার অংশীদার। A মোট মূলধনের \( \frac{1}{4} \) অংশ \(15\) মাসের জন্য বিনিয়োগ করে এবং B লাভের \( \frac{2}{3} \) অংশ পায়। B-এর অর্থ কত মাস ব্যবহার করা হয়েছিল?

Profit is divided in the ratio of investment × time.

A's share : B's share =

\[ \frac{1}{3} : \frac{2}{3} \]

\[ = 1 : 2 \]

Let total capital = \(1\).

A invests \( \frac{1}{4} \).

So B invests \( \frac{3}{4} \).

Let B's money be used for \(x\) months.

According to rule:

\[ \frac{1}{4} \times 15 : \frac{3}{4} \times x = 1 : 2 \]

\[ \frac{15}{4} : \frac{3x}{4} = 1 : 2 \]

\[ 15 : 3x = 1 : 2 \]

\[ \frac{15}{3x} = \frac{1}{2} \]

\[ 30 = 3x \]

\[ x = 10 \]

So, B's money was used for \(10\) months.

Correct Answer: C