In the year 2019, R has invested for six months only while Q and P invested for the whole year. Find the total profit earned during the year 2019.

Steps to Solve

1. Identify Profit Data for 2019 From the table, we know the profit earned by P in 2019 is $6400$, while the profits for R and Q are missing.

2. Understand R's Investment Duration R invested for $6$ months in the year 2019, while P and Q are assumed to have invested the entire year (12 months).

3. Apply the Profit Sharing Rule The profit is generally shared in the ratio of the product of capital invested and the duration of investment.

   Let:

   • R's capital = $R_c$
   • P's capital = $6400$ (since this is their profit for the year),
   • Q's capital = $Q_c$

   Since R only invested for $6$ months:

   • Effective investment ratio for R = $R_c \times 6$
   • Effective investment ratio for P = $6400 \times 12$

4. Determine Capital of P Keeping the investments proportional, we equate the total profit distribution:

   $$ P_{\text{total}} = P + R + Q = 6400 + R + Q = 19200 $$

   Here, we can substitute R's profit contribution based on the ratios.

5. Calculate Total Profit and Determine R's Contribution in 2019 Set up the profit equations:

   $$ R_{\text{profit}} = \text{Proportion of Profit} \times (6400 + R + Q) $$

   Calculate R's share to find the final profit earned by R:

   $$ \text{Total Profit} = 6400 + \text{R's share} $$