Clara and Daniel launched a joint venture, contributing capital in the ratio of 2:5. Out of the total profit of $6,900, Clara received $2,400 as her dividend. If the difference between the investment durations of Clara and Daniel is 2 months, for how many months did Clara keep her money invested?

The correct answer is 8.

Step 1: Understand the Capital Ratio
Clara and Daniel contributed capital in the ratio of 2 : 5.
Let Clara's capital contribution be $2x$ and Daniel's capital contribution be $5x$.

Step 2: Calculate Daniel's Profit Share
The total profit from the joint venture is $6,900, and Clara received $2,400 as her dividend.

$\text{Daniel's profit}=\text{Total profit}-\text{Clara's profit}$

$\text{Daniel's profit}=6900-2400=4500$

Now, calculate the ratio of Clara's profit to Daniel's profit:

$\frac{\text{Clara's profit}}{\text{Daniel's profit}}=\frac{2400}{4500}=\frac{8}{15}$

Step 3: Relate Profit Ratio to Capital and Time
The share of profit in a partnership is directly proportional to the product of capital invested and the duration of investment.
Let $T_c$ be the duration of Clara's investment in months, and $T_d$ be the duration of Daniel's investment in months.

$\frac{\text{Clara's profit}}{\text{Daniel's profit}}=\frac{\text{Clara's capital}\times T_c}{\text{Daniel's capital}\times T_d}$

Substitute the known values into the equation:

$\frac{8}{15}=\frac{2x\times T_c}{5x\times T_d}$

Cancel out $x$ from the numerator and denominator:

$\frac{8}{15}=\frac{2T_c}{5T_d}$

Solve for the ratio of their investment durations $\frac{T_c}{T_d}$:

$\frac{T_c}{T_d}=\frac{8}{15}\times \frac{5}{2}=\frac{4}{3}$

This shows that Clara's investment duration $T_c$ is longer than Daniel's investment duration $T_d$, with $T_c=\frac{4}{3}{T}_d$.

Step 4: Use the Difference in Investment Durations
We are given that the difference between the investment durations of Clara and Daniel is 2 months:

$T_c-T_d=2$

Substitute $T_c=\frac{4}{3}{T}_d$ into the difference equation:

$\frac{4}{3}{T}_d-T_d=2$

$\frac{1}{3}{T}_d=2$

$T_d=6\text{months}$

Step 5: Calculate Clara's Investment Duration
Now calculate Clara's investment duration $T_c$:

$T_c=T_d+2=6+2=8\text{months}$

Clara kept her money invested for 8 months.