Three persons A, B and C together undertake to complete a piece of work for Rs 1600. A can complete the work alone in 6 days, B alone in 15 days and C alone in 24 days respectively. If they complete the work with the help of person D in 3 days, then find the wage of person D?

Options

A. A)Rs 250
B. B)Rs 350
C. C)Rs 210
D. D)Rs 280

Answer

To solve the problem step by step, we need to determine the individual work rates of A, B, and C, and then find out how much work D contributes when they all work together. Finally, we will calculate D's wage based on the total payment for the work done.

### Step-by-Step Solution:

1. **Calculate the work rate of A, B, and C:**
- A can complete the work in 6 days, so A's work rate is:
\[ \text{Work rate of A} = \frac{1 \text{ work}}{6 \text{ days}} = \frac{1}{6} \text{ work/day} \]
- B can complete the work in 15 days, so B's work rate is:
\[ \text{Work rate of B} = \frac{1 \text{ work}}{15 \text{ days}} = \frac{1}{15} \text{ work/day} \]
- C can complete the work in 24 days, so C's work rate is:
\[ \text{Work rate of C} = \frac{1 \text{ work}}{24 \text{ days}} = \frac{1}{24} \text{ work/day} \]

2. **Find the combined work rate of A, B, and C:**
- To find the combined work rate of A, B, and C, we add their individual work rates:
\[ \text{Combined work rate of A, B, C} = \frac{1}{6} + \frac{1}{15} + \frac{1}{24} \]
- To add these fractions, we need a common denominator. The LCM of 6, 15, and 24 is 120.
- Convert each fraction:
\[ \frac{1}{6} = \frac{20}{120}, \quad \frac{1}{15} = \frac{8}{120}, \quad \frac{1}{24} = \frac{5}{120} \]
- Now, add them:
\[ \frac{20}{120} + \frac{8}{120} + \frac{5}{120} = \frac{33}{120} \]

3. **Calculate the total work done by A, B, C, and D in 3 days:**
- The total work done by A, B, and C in one day is \(\frac{33}{120}\). Therefore, in 3 days:
\[ \text{Work done by A, B, C in 3 days} = 3 \times \frac{33}{120} = \frac{99}{120} = \frac{33}{40} \]
- Since they completed the work with the help of D, the total work done in 3 days is 1 (the whole work):
\[ \text{Work done by D in 3 days} = 1 - \frac{33}{40} = \frac{7}{40} \]

4. **Calculate the work rate of D:**
- The work rate of D can be found by dividing the work done by D in 3 days by 3:
\[ \text{Work rate of D} = \frac{7/40}{3} = \frac{7}{120} \text{ work/day} \]

5. **Determine the total efficiency of A, B, C, and D:**
- The total efficiency when A, B, C, and D work together is:
\[ \text{Total efficiency} = \frac{33}{120} + \frac{7}{120} = \frac{40}{120} = \frac{1}{3} \text{ work/day} \]

6. **Calculate the wage of D:**
- The total payment for the work is Rs 1600. Since the total efficiency is 40 units (from the work done in 3 days), we can find the value of 1 unit of work:
\[ \text{Value of 1 unit} = \frac{1600}{40} = 40 \text{ Rs} \]
- D's share of work is 7 units (from the work done in 3 days), so D's wage is:
\[ \text{D's wage} = 7 \times 40 = 280 \text{ Rs} \]

### Final Answer:
D's wage is Rs 280.