A, B and C can do a piece of work in 20, 24 and 30 days, respectively. They undertook to do the piece of work for Rs. 5400. They begin the work together but B left 2 days before the completion of work and c left 5 days before the completion of work. The share of A from the assured money is

Options

A. Rs.2700
B. Rs. 540
C. Rs. 1800
D. Rs. 600

Answer

To solve the problem step by step, we will follow the outlined approach in the video transcript.

### Step 1: Determine the Work Rates of A, B, and C
- A can complete the work in 20 days, so A's work rate is \( \frac{1}{20} \) of the work per day.
- B can complete the work in 24 days, so B's work rate is \( \frac{1}{24} \) of the work per day.
- C can complete the work in 30 days, so C's work rate is \( \frac{1}{30} \) of the work per day.

### Step 2: Find the Total Work
To find a common measure for the work done, we can take the least common multiple (LCM) of the days:
- The LCM of 20, 24, and 30 is 120.
- Therefore, the total work can be assumed to be 120 units.

### Step 3: Calculate Individual Work Contributions
Now, we calculate the work done by each person in terms of units:
- A's work in 120 units: \( \frac{120}{20} = 6 \) units per day.
- B's work in 120 units: \( \frac{120}{24} = 5 \) units per day.
- C's work in 120 units: \( \frac{120}{30} = 4 \) units per day.

### Step 4: Calculate Total Work Done Together
The combined work done by A, B, and C in one day is:
- Total work rate = \( 6 + 5 + 4 = 15 \) units per day.

### Step 5: Determine the Total Time Worked
Let’s assume they worked together for \( x \) days. Since B left 2 days before the completion and C left 5 days before the completion, we can express the total work done as:
- A worked for \( x \) days.
- B worked for \( x - 2 \) days.
- C worked for \( x - 5 \) days.

The total work can be expressed as:
\[ 6x + 5(x - 2) + 4(x - 5) = 120 \]

### Step 6: Simplify the Equation
Expanding the equation:
\[ 6x + 5x - 10 + 4x - 20 = 120 \]
Combining like terms:
\[ 15x - 30 = 120 \]
Adding 30 to both sides:
\[ 15x = 150 \]
Dividing by 15:
\[ x = 10 \]

### Step 7: Calculate Work Done by Each Person
Now we can calculate how much work each person did:
- A's work: \( 6 \times 10 = 60 \) units.
- B's work: \( 5 \times 8 = 40 \) units (B worked for 8 days).
- C's work: \( 4 \times 5 = 20 \) units (C worked for 5 days).

### Step 8: Calculate the Share of A
The total work is 120 units, and the payment is Rs. 5400. The share of A can be calculated as follows:
\[ \text{Share of A} = \frac{\text{Work done by A}}{\text{Total work}} \times \text{Total payment} \]
\[ \text{Share of A} = \frac{60}{120} \times 5400 = \frac{1}{2} \times 5400 = 2700 \]

### Final Answer
Thus, the share of A from the assured money is Rs. 2700.

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