A, B and C working together can finish a piece of work in 32 days. They worked for 24 days and after that the C left. Rest of the work is completed by A and B in 28 days. Work done by A in 1 day is equal to work done by B in 2 days. In how many days working alone each one will finish the work?

Answer

To solve the problem step by step, we will first determine the work done by A, B, and C together, then find their individual efficiencies, and finally calculate how many days each one would take to finish the work alone.

### Step 1: Determine the total work done
A, B, and C together can finish the work in 32 days. Therefore, the total work can be represented in terms of work units:
\[ \text{Total Work} = \text{Work Rate} \times \text{Time} = \frac{1}{32} \times 32 = 1 \text{ unit of work} \]

### Step 2: Calculate the work done in 24 days
In 24 days, A, B, and C will complete:
\[ \text{Work done in 24 days} = \frac{1}{32} \times 24 = \frac{24}{32} = \frac{3}{4} \text{ units of work} \]

### Step 3: Calculate the remaining work
The remaining work after 24 days is:
\[ \text{Remaining Work} = 1 - \frac{3}{4} = \frac{1}{4} \text{ units of work} \]

### Step 4: Work done by A and B in 28 days
A and B complete the remaining work in 28 days. Therefore, their combined work rate can be calculated as:
\[ \text{Work Rate of A and B} = \frac{1}{4} \div 28 = \frac{1}{112} \text{ units of work per day} \]

### Step 5: Establish the relationship between A and B
We know that the work done by A in 1 day is equal to the work done by B in 2 days. Therefore, if A's work in one day is \( a \), then B's work in one day is \( \frac{a}{2} \).

Let the work done by A in one day be \( a \) and the work done by B in one day be \( b \). Thus:
\[ b = \frac{a}{2} \]

### Step 6: Combine the efficiencies of A and B
The combined work done by A and B in one day is:
\[ a + b = a + \frac{a}{2} = \frac{3a}{2} \]
We know from step 4 that:
\[ \frac{3a}{2} = \frac{1}{112} \]
Solving for \( a \):
\[ 3a = \frac{2}{112} = \frac{1}{56} \implies a = \frac{1}{168} \]
Thus, A's work rate is \( \frac{1}{168} \) units of work per day.

### Step 7: Calculate B's work rate
Using \( b = \frac{a}{2} \):
\[ b = \frac{1/168}{2} = \frac{1}{336} \]
So, B's work rate is \( \frac{1}{336} \) units of work per day.

### Step 8: Calculate C's work rate
We already know the combined work rate of A, B, and C:
\[ \text{Work Rate of A + B + C} = \frac{1}{32} \]
Now substituting A and B's work rates:
\[ \frac{1}{168} + \frac{1}{336} + c = \frac{1}{32} \]
Finding a common denominator for \( \frac{1}{168} \) and \( \frac{1}{336} \) (which is 336):
\[ \frac{2}{336} + \frac{1}{336} + c = \frac{1}{32} \]
\[ \frac{3}{336} + c = \frac{1}{32} \]
Converting \( \frac{1}{32} \) to a fraction with a denominator of 336:
\[ \frac{1}{32} = \frac{10.5}{336} \]
Thus:
\[ c = \frac{10.5}{336} - \frac{3}{336} = \frac{7.5}{336} = \frac{15}{672} \]

### Step 9: Calculate the time taken by each worker to finish the work alone
- For A:
\[ \text{Time taken by A} = \frac{1}{\frac{1}{168}} = 168 \text{ days} \]
- For B:
\[ \text{Time taken by B} = \frac{1}{\frac{1}{336}} = 336 \text{ days} \]
- For C:
\[ \text{Time taken by C} = \frac{1}{\frac{15}{672}} = \frac{672}{15} = 44.8 \text{ days} \]

### Final Answer:
- A can finish the work alone in **168 days**.
- B can finish the work alone in **336 days**.
- C can finish the work alone in **44.8 days**.