To solve this problem, let's break down the work rates of A, B, and C and then find C's share of the payment.

Step 1: Find the individual work rates of A and B

• A can complete the work in 48 days, so A's work rate is: $\text{Rate of A} = \frac{1}{48} \text{ of the work per day.}$
• B can complete the work in 72 days, so B's work rate is: $\text{Rate of B} = \frac{1}{72} \text{ of the work per day.}$

Step 2: Find the combined rate of A, B, and C

Together, A, B, and C complete the work in 24 days, so their combined rate is: $\text{Rate of A, B, and C} = \frac{1}{24} \text{ of the work per day.}$

Step 3: Find C's work rate

Let the rate of C be denoted by $\text{Rate of C} = \frac{1}{x}$. Since the combined rate of A, B, and C is the sum of their individual rates: $\frac{1}{48} + \frac{1}{72} + \frac{1}{x} = \frac{1}{24}.$ Now, let's solve for $x$.

First, find the least common denominator (LCD) of the fractions on the left side. The LCD of 48, 72, and 24 is 144. Rewrite each fraction with the denominator 144: $\frac{1}{48} = \frac{3}{144}, \quad \frac{1}{72} = \frac{2}{144}, \quad \frac{1}{24} = \frac{6}{144}.$ Substitute these into the equation: $\frac{3}{144} + \frac{2}{144} + \frac{1}{x} = \frac{6}{144}.$ Simplifying the equation: $\frac{5}{144} + \frac{1}{x} = \frac{6}{144}.$ Subtract $\frac{5}{144}$ from both sides: $\frac{1}{x} = \frac{6}{144} - \frac{5}{144} = \frac{1}{144}.$ Thus, $x = 144$, so C can complete the work in 144 days.

Step 4: Find the share of C in the payment

The amount of work completed by each person can be calculated by their work rate relative to the total work. The total work done is 1 unit of work.

• A's share of the work: $\frac{1}{48} \times 24 = \frac{1}{2}$
• B's share of the work: $\frac{1}{72} \times 24 = \frac{1}{3}$
• C's share of the work: $\frac{1}{144} \times 24 = \frac{1}{6}$

Step 5: Calculate C's payment

The total payment is $3000. Since C completed $\frac{1}{6}$ of the work, C's share of the payment is: $\text{C's payment} = \frac{1}{6} \times 3000 = 500.$

Thus, C should receive $500 for the work done.

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Would you like further details or have any questions about this? Here are some related questions:

1. How can we calculate the share of payment for other workers when they complete tasks at different rates?
2. What would happen if A, B, and C worked together for a different number of days? How would that affect the shares?
3. How would you calculate the total work if only two workers are involved?
4. What is the importance of finding the combined work rate when multiple workers are involved in a task?
5. How do you handle situations where the workers’ rates are not in simple numbers (like 48, 72, etc.)?

Tip: When dealing with rates, always find a common denominator or use the least common multiple to simplify the calculation process.