A and B can do a piece of work in 30 days while B and C can do the same work in 24 days and and A in 20 days. They all work together for 10 days when B and Cleave. How many days more will A take to finish the work ?

Options

A. 18 days
B. 24 days
C. 30 days
D. 36 days

Answer

To solve the problem step by step, we will determine the work efficiencies of A, B, and C based on the information provided and then calculate how long A will take to finish the remaining work after B and C leave.

### Step 1: Determine the work efficiencies of A, B, and C

1. **A and B can complete the work in 30 days.**
- Work done by A and B in one day = 1/30 of the work.

2. **B and C can complete the work in 24 days.**
- Work done by B and C in one day = 1/24 of the work.

3. **C and A can complete the work in 20 days.**
- Work done by C and A in one day = 1/20 of the work.

### Step 2: Set up equations based on the efficiencies

Let:
- Efficiency of A = a
- Efficiency of B = b
- Efficiency of C = c

From the above information, we can set up the following equations:
1. \( a + b = \frac{1}{30} \) (1)
2. \( b + c = \frac{1}{24} \) (2)
3. \( c + a = \frac{1}{20} \) (3)

### Step 3: Solve the equations

To find the individual efficiencies, we can add all three equations:

\[ (a + b) + (b + c) + (c + a) = \frac{1}{30} + \frac{1}{24} + \frac{1}{20} \]

This simplifies to:

\[ 2a + 2b + 2c = \frac{1}{30} + \frac{1}{24} + \frac{1}{20} \]

Now, calculate the right side using LCM:

- LCM of 30, 24, and 20 is 120.
- Convert each fraction:
- \( \frac{1}{30} = \frac{4}{120} \)
- \( \frac{1}{24} = \frac{5}{120} \)
- \( \frac{1}{20} = \frac{6}{120} \)

Adding these gives:

\[ \frac{4 + 5 + 6}{120} = \frac{15}{120} = \frac{1}{8} \]

Thus, we have:

\[ 2(a + b + c) = \frac{1}{8} \]

Dividing by 2:

\[ a + b + c = \frac{1}{16} \]

### Step 4: Find individual efficiencies

Now we can find the individual efficiencies:

1. From equation (1): \( b = \frac{1}{30} - a \)
2. Substitute \( b \) in equation (2):
- \( \frac{1}{30} - a + c = \frac{1}{24} \)
- Rearranging gives \( c = \frac{1}{24} - \frac{1}{30} + a \)

3. Substitute \( c \) in equation (3):
- \( a + \left(\frac{1}{24} - \frac{1}{30} + a\right) = \frac{1}{20} \)
- Solve for \( a \).

After solving, we find:
- \( a = \frac{1}{40} \)
- \( b = \frac{1}{60} \)
- \( c = \frac{1}{30} \)

### Step 5: Calculate the combined efficiency

Now, we can find the combined efficiency of A, B, and C:

\[ a + b + c = \frac{1}{40} + \frac{1}{60} + \frac{1}{30} = \frac{3 + 2 + 4}{120} = \frac{9}{120} = \frac{3}{40} \]

### Step 6: Work done in 10 days

If they work together for 10 days:

\[ \text{Work done in 10 days} = 10 \times \frac{3}{40} = \frac{30}{40} = \frac{3}{4} \]

### Step 7: Remaining work

Total work = 1 (whole work), so remaining work:

\[ 1 - \frac{3}{4} = \frac{1}{4} \]

### Step 8: A's efficiency

We already calculated A's efficiency as \( \frac{1}{40} \).

### Step 9: Time taken by A to finish the remaining work

To find how many days A will take to finish the remaining \( \frac{1}{4} \):

\[ \text{Time} = \frac{\text{Remaining Work}}{\text{A's Efficiency}} = \frac{\frac{1}{4}}{\frac{1}{40}} = \frac{40}{4} = 10 \text{ days} \]

### Final Answer

A will take **10 more days** to finish the work.

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