A, B and C can finish a piece of work in 20 days, 25 days and 30 days respectively. They started working together. After how many days A should leave the work so that total work will be complete in 12 days?

Answer

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

### Step 1: Determine the work done by A, B, and C in one day.
- A can finish the work in 20 days, so A's work per day = \( \frac{1}{20} \)
- B can finish the work in 25 days, so B's work per day = \( \frac{1}{25} \)
- C can finish the work in 30 days, so C's work per day = \( \frac{1}{30} \)

### Step 2: Calculate the total work done by A, B, and C together in one day.
To find the total work done by A, B, and C together in one day, we add their efficiencies:
\[ \text{Total work per day} = \frac{1}{20} + \frac{1}{25} + \frac{1}{30} \]

To add these fractions, we first find the least common multiple (LCM) of the denominators (20, 25, and 30).
- LCM of 20, 25, and 30 is 300.

Now, we convert each fraction to have a denominator of 300:
- A's work: \( \frac{1}{20} = \frac{15}{300} \)
- B's work: \( \frac{1}{25} = \frac{12}{300} \)
- C's work: \( \frac{1}{30} = \frac{10}{300} \)

Now, adding these gives:
\[ \text{Total work per day} = \frac{15 + 12 + 10}{300} = \frac{37}{300} \]

### Step 3: Calculate the total work done by B and C in 12 days.
We need to find out how much work B and C can do in 12 days:
\[ \text{Work done by B and C in 12 days} = (B + C) \times 12 \]
First, we find the combined work of B and C:
\[ B + C = \frac{12}{300} + \frac{10}{300} = \frac{22}{300} \]
Now, work done by B and C in 12 days:
\[ \text{Work done by B and C in 12 days} = \frac{22}{300} \times 12 = \frac{264}{300} = \frac{44}{50} = \frac{22}{25} \]

### Step 4: Calculate the remaining work.
The total work is 1 (or 300 units). Therefore, the remaining work that A needs to complete is:
\[ \text{Remaining work} = 1 - \frac{22}{25} = \frac{3}{25} \]

### Step 5: Determine how many days A needs to work to finish the remaining work.
A's work per day is \( \frac{15}{300} = \frac{1}{20} \). To find out how many days A needs to finish the remaining work:
\[ \text{Days A needs to work} = \text{Remaining work} \div \text{A's work per day} = \frac{3}{25} \div \frac{1}{20} = \frac{3}{25} \times 20 = \frac{60}{25} = 2.4 \text{ days} \]

### Step 6: Calculate how many days A should leave the work.
Since they are working together for 12 days, if A works for 2.4 days, then:
\[ \text{Days A should leave} = 12 - 2.4 = 9.6 \text{ days} \]

### Final Answer:
A should leave the work after approximately 9.6 days.

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