A and B undertook to do a piece of work for 4500. A alone could do it in 8 days and B alone in 12 days. With the assistance of C they finished the work in 4 days. Then C's share of the money is

Options

A. Rs 2250
B. Rs 1500
C. Rs 750
D. Rs 375

Answer

To solve the problem step by step, we will determine how much work A, B, and C can do individually and then find out C's share of the total payment.

### Step 1: Calculate the work rates of A and B
- A can complete the work in 8 days, so A's work rate is:
\[ \text{Work rate of A} = \frac{1}{8} \text{ (work per day)} \]
- B can complete the work in 12 days, so B's work rate is:
\[ \text{Work rate of B} = \frac{1}{12} \text{ (work per day)} \]

### Step 2: Calculate the combined work rate of A and B
To find the combined work rate of A and B, we add their individual work rates:
\[ \text{Combined work rate of A and B} = \frac{1}{8} + \frac{1}{12} \]
To add these fractions, we need a common denominator. The least common multiple (LCM) of 8 and 12 is 24.
\[ \frac{1}{8} = \frac{3}{24}, \quad \frac{1}{12} = \frac{2}{24} \]
Thus,
\[ \text{Combined work rate of A and B} = \frac{3}{24} + \frac{2}{24} = \frac{5}{24} \]

### Step 3: Calculate the combined work rate of A, B, and C
With the assistance of C, they finished the work in 4 days, so their combined work rate is:
\[ \text{Combined work rate of A, B, and C} = \frac{1}{4} \]

### Step 4: Find C's work rate
Now, we can find C's work rate by subtracting the combined work rate of A and B from the combined work rate of A, B, and C:
\[ \text{Work rate of C} = \frac{1}{4} - \frac{5}{24} \]
To perform this subtraction, we need a common denominator, which is 24:
\[ \frac{1}{4} = \frac{6}{24} \]
Thus,
\[ \text{Work rate of C} = \frac{6}{24} - \frac{5}{24} = \frac{1}{24} \]

### Step 5: Calculate the time taken by C to complete the work alone
Since C's work rate is \(\frac{1}{24}\), it means C can complete the work alone in 24 days.

### Step 6: Determine the ratio of work done by A, B, and C
Now we can find the ratio of work done by A, B, and C:
- Work done by A in 24 days:
\[ \text{Work by A} = \frac{24}{8} = 3 \text{ units} \]
- Work done by B in 24 days:
\[ \text{Work by B} = \frac{24}{12} = 2 \text{ units} \]
- Work done by C in 24 days:
\[ \text{Work by C} = \frac{24}{24} = 1 \text{ unit} \]
Thus, the ratio of work done by A, B, and C is \(3:2:1\).

### Step 7: Calculate the total parts and C's share
The total parts of work done by A, B, and C is:
\[ 3 + 2 + 1 = 6 \text{ parts} \]
C's share of the total payment of Rs. 4500 is:
\[ \text{C's share} = \frac{1}{6} \times 4500 = 750 \text{ Rs.} \]

### Final Answer
C's share of the money is Rs. 750.

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