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 along. What is the share of S ( in Rs. ) from total money ?

Options

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

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Determine the total work
The total work can be calculated using the Least Common Multiple (LCM) of the days taken by S, T, and U to complete the work.

- S can complete the work in 40 days.
- T can complete the work in 48 days.
- U can complete the work in 60 days.

To find the LCM of 40, 48, and 60:
- Prime factorization:
- 40 = 2^3 * 5
- 48 = 2^4 * 3
- 60 = 2^2 * 3 * 5

The LCM will take the highest power of each prime factor:
- LCM = 2^4 * 3^1 * 5^1 = 240

Thus, the total work is 240 units.

### Step 2: Calculate the work done by each person in one day
Now, we will calculate how much work each person can do in one day.

- Work done by S in one day = Total work / Days taken by S = 240 / 40 = 6 units
- Work done by T in one day = Total work / Days taken by T = 240 / 48 = 5 units
- Work done by U in one day = Total work / Days taken by U = 240 / 60 = 4 units

### Step 3: Calculate the combined work done in one day
The combined work done by S, T, and U in one day is:
- Total work done in one day = Work by S + Work by T + Work by U = 6 + 5 + 4 = 15 units

### Step 4: Determine the total time taken to complete the work
Let X be the total number of days taken to complete the work.

- S worked for X days.
- T worked for (X - 2) days (since T left 2 days before completion).
- U worked for (X - 5) days (since U left 5 days before completion).

The equation for the total work done is:
\[ 6X + 5(X - 2) + 4(X - 5) = 240 \]

Expanding this:
\[ 6X + 5X - 10 + 4X - 20 = 240 \]
\[ 15X - 30 = 240 \]
\[ 15X = 270 \]
\[ X = 18 \]

### Step 5: Calculate the work done by each person
Now we can calculate how much work each person completed:
- Work done by S = 6 * 18 = 108 units
- Work done by T = 5 * (18 - 2) = 5 * 16 = 80 units
- Work done by U = 4 * (18 - 5) = 4 * 13 = 52 units

### Step 6: Calculate the total work done
Total work done = Work by S + Work by T + Work by U = 108 + 80 + 52 = 240 units (which confirms our calculations).

### Step 7: Calculate the share of S from the total money
The total money received is Rs. 10800. The share of each person is proportional to the work done.

- Total work done = 240 units
- Share of S = (Work done by S / Total work) * Total money
\[ \text{Share of S} = \left(\frac{108}{240}\right) \times 10800 \]
\[ \text{Share of S} = \frac{108 \times 10800}{240} \]
\[ \text{Share of S} = \frac{1166400}{240} = 4860 \]

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