Calculating Work Rates and Shares

First, determine the individual work rates of A and B, and their combined work rate with C.

• A's work rate: A finishes the job in 12 days, so A's rate is $\frac{1}{12}$ of the job per day.
• B's work rate: B finishes the job in 15 days, so B's rate is $\frac{1}{15}$ of the job per day.
• Combined work rate (A, B, and C): They finish the job in 5 days, so their combined rate is $\frac{1}{5}$ of the job per day.

Finding C's Work Rate

Calculate C's individual work rate by subtracting the combined rate of A and B from the combined rate of A, B, and C.

1. Calculate the combined rate of A and B: $ \text{Rate}_{A+B} = \frac{1}{12} + \frac{1}{15} $ Find a common denominator (60): $ \text{Rate}_{A+B} = \frac{5}{60} + \frac{4}{60} = \frac{9}{60} = \frac{3}{20} $ job per day.
2. Calculate C's rate: $ \text{Rate}_{C} = \text{Rate}_{A+B+C} - \text{Rate}_{A+B} $ $ \text{Rate}_{C} = \frac{1}{5} - \frac{3}{20} $ Find a common denominator (20): $ \text{Rate}_{C} = \frac{4}{20} - \frac{3}{20} = \frac{1}{20} $ job per day.

Determining Share Based on Work Rate

The payment is divided proportionally to the work rates. Find the ratio of work rates.

1. Ratio of work rates (A : B : C) = $ \frac{1}{12} : \frac{1}{15} : \frac{1}{20} $.
2. To simplify the ratio, multiply by the LCM of 12, 15, and 20, which is 60: $ (\frac{1}{12} \times 60) : (\frac{1}{15} \times 60) : (\frac{1}{20} \times 60) $ $ 5 : 4 : 3 $
3. Total parts in the ratio = $ 5 + 4 + 3 = 12 $.
4. C's share is $\frac{3}{12}$ (or $\frac{1}{4}$) of the total payment.
5. Calculate C's share: $ \text{C's Share} = \frac{3}{12} \times ₹2,880 $ $ \text{C's Share} = \frac{1}{4} \times ₹2,880 $ $ \text{C's Share} = ₹720 $

Therefore, C's share of the payment is ₹720.