Directions: Solve the question below.

Clara and David started a commercial partnership such that the sum invested by Clara is $\frac{3}{7}$ of the total sum invested by both of them combined, and Clara invested her capital for 8 months. The profit received by David is $\frac{1}{3}$ of the total profit received by Clara and David together. Find the duration (in months) for which David invested his capital.

The correct answer is 3.

Step 1: Find the ratio of investments between Clara and David.

Let the total investment made by Clara and David combined be $I$.

It is given that the sum invested by Clara is $\frac{3}{7}$ of the total sum invested by both of them combined.

Therefore, Clara's investment ($I_{\text{Clara}}$) is:

$I_{\text{Clara}}=\frac{3}{7}I$

David's investment ($I_{\text{David}}$) is the remaining portion of the total investment:

$I_{\text{David}}=I-\frac{3}{7}I=\frac{4}{7}I$

Thus, the ratio of Clara's investment to David's investment is:

$I_{\text{Clara}}:I_{\text{David}}=\frac{3}{7}I:\frac{4}{7}I=3:4$

Step 2: Find the ratio of profits received by Clara and David.

Let the total profit received by Clara and David together be $P$.

The profit received by David is given as $\frac{1}{3}$ of the total profit:

$P_{\text{David}}=\frac{1}{3}P$

The profit received by Clara ($P_{\text{Clara}}$) is the remaining share of the total profit:

$P_{\text{Clara}}=P-\frac{1}{3}P=\frac{2}{3}P$

Thus, the ratio of profit received by Clara to that received by David is:

$P_{\text{Clara}}:P_{\text{David}}=\frac{2}{3}P:\frac{1}{3}P=2:1$

Step 3: Determine the duration of David's investment.

In a partnership, profit is distributed in proportion to the product of the investment and the time period for which it is invested:

$\frac{P_{\text{Clara}}}{P_{\text{David}}}=\frac{I_{\text{Clara}}\times T_{\text{Clara}}}{I_{\text{David}}\times T_{\text{David}}}$

We are given that Clara invested her capital for 8 months ($T_{\text{Clara}}=8$). Let $T_{\text{David}}$ be the duration of David's investment in months.

Substituting the investment ratio ($3:4$), profit ratio ($2:1$), and Clara's time ($8$ months) into the formula:

$\frac{2}{1}=\frac{3\times 8}{4\times T_{\text{David}}}$

Simplify the fraction on the right side:

$\frac{2}{1}=\frac{24}{4T_{\text{David}}}$

$\frac{2}{1}=\frac{6}{T_{\text{David}}}$

Cross-multiply to solve for $T_{\text{David}}$:

$2\times T_{\text{David}}=6$

$T_{\text{David}}=\frac{6}{2}=3$

Therefore, David invested his capital for 3 months.