S, T and U can complete a work In 40, 48 and 60 days respectively. They received Rs. 10800 to complete the work. They begin the work together but T left 2 days before the completion of the work and U left 5 days before the completion of the work. S has completed the remaining work alone. What is the share of Sin Rs.) from total money?

Options

A. 4000
B. 4320
C. 4500
D. 4860

Answer

To solve the problem step by step, we will first determine the work done by each person and then calculate their respective shares from the total money.

### Step 1: Determine the work done by S, T, and U
- S can complete the work in 40 days, so S's work rate is \( \frac{1}{40} \) of the work per day.
- T can complete the work in 48 days, so T's work rate is \( \frac{1}{48} \) of the work per day.
- U can complete the work in 60 days, so U's work rate is \( \frac{1}{60} \) of the work per day.

### Step 2: Find the LCM of the days to calculate work in units
To find a common unit of work, we will calculate the LCM of 40, 48, and 60.
- The LCM of 40, 48, and 60 is 240.
- Therefore, the total work can be considered as 240 units.

### Step 3: Calculate the work done by each person in one day
- S's work in one day = \( \frac{240}{40} = 6 \) units
- T's work in one day = \( \frac{240}{48} = 5 \) units
- U's work in one day = \( \frac{240}{60} = 4 \) units

### Step 4: Calculate the total work done by S, T, and U together
- Combined work done in one day by S, T, and U = \( 6 + 5 + 4 = 15 \) units.

### Step 5: Determine the total time taken to complete the work
Let the total number of days taken to complete the work be \( x \).
- T left 2 days before completion, so T worked for \( x - 2 \) days.
- U left 5 days before completion, so U worked for \( x - 5 \) days.
- S worked for all \( x \) days.

The total work equation can be set up as:
\[ 6x + 5(x - 2) + 4(x - 5) = 240 \]

### Step 6: Simplify the equation
Expanding the equation:
\[ 6x + 5x - 10 + 4x - 20 = 240 \]
Combining like terms:
\[ 15x - 30 = 240 \]
Adding 30 to both sides:
\[ 15x = 270 \]
Dividing by 15:
\[ x = 18 \]

### Step 7: Calculate the work done by each person
- Work done by S in 18 days = \( 6 \times 18 = 108 \) units.
- Work done by T in \( 18 - 2 = 16 \) days = \( 5 \times 16 = 80 \) units.
- Work done by U in \( 18 - 5 = 13 \) days = \( 4 \times 13 = 52 \) units.

### Step 8: Verify total work done
Total work done = \( 108 + 80 + 52 = 240 \) units (which matches the total work).

### Step 9: Calculate the share of each person
The total payment is Rs. 10800. The share of each person is proportional to the work done:
- Total units of work = 240
- Value of 1 unit = \( \frac{10800}{240} = 45 \) Rs.

### Step 10: Calculate S's share
- S's share = Work done by S × Value of 1 unit
\[ S's \, share = 108 \times 45 = 4860 \, Rs. \]

### Final Answer:
The share of S from the total money is Rs. 4860.

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