To solve the problem step by step, we will first determine the work rates of A, B, and C, and then find out how long it will take for A, B, and C to complete the work together.
### Step 1: Calculate the total work in terms of work units.
A can do \( \frac{1}{3} \) of the work in 30 days. Therefore, the total work can be considered as 1 unit of work.
If A does \( \frac{1}{3} \) of the work in 30 days, then the total work (W) can be calculated as:
\[
W = 3 \times 30 = 90 \text{ days of work}
\]
### Step 2: Calculate the work rate of A.
A's work rate is:
\[
\text{Work rate of A} = \frac{1}{3} \text{ of work} / 30 \text{ days} = \frac{1}{90} \text{ of work per day}
\]
### Step 3: Calculate the work rate of B.
B can do \( \frac{2}{5} \) of the work in 24 days. Therefore, B's work rate is:
\[
\text{Work rate of B} = \frac{2}{5} \text{ of work} / 24 \text{ days} = \frac{2}{120} = \frac{1}{60} \text{ of work per day}
\]
### Step 4: Calculate the combined work rate of A and B.
The combined work rate of A and B is:
\[
\text{Combined work rate of A and B} = \frac{1}{90} + \frac{1}{60}
\]
To add these fractions, find a common denominator, which is 180:
\[
\frac{1}{90} = \frac{2}{180}, \quad \frac{1}{60} = \frac{3}{180}
\]
Thus,
\[
\text{Combined work rate of A and B} = \frac{2}{180} + \frac{3}{180} = \frac{5}{180} = \frac{1}{36} \text{ of work per day}
\]
### Step 5: Calculate the work done by A and B in 20 days.
In 20 days, A and B together will complete:
\[
\text{Work done by A and B in 20 days} = 20 \times \frac{1}{36} = \frac{20}{36} = \frac{5}{9} \text{ of the work}
\]
### Step 6: Calculate the remaining work.
The remaining work after A and B have worked together for 20 days is:
\[
\text{Remaining work} = 1 - \frac{5}{9} = \frac{4}{9} \text{ of the work}
\]
### Step 7: Calculate the work rate of C.
Let’s assume C can complete the remaining work in 8 days. Therefore, C's work rate is:
\[
\text{Work rate of C} = \frac{4}{9} \text{ of work} / 8 \text{ days} = \frac{4}{72} = \frac{1}{18} \text{ of work per day}
\]
### Step 8: Calculate the combined work rate of A, B, and C.
Now, we can find the combined work rate of A, B, and C:
\[
\text{Combined work rate of A, B, and C} = \frac{1}{90} + \frac{1}{60} + \frac{1}{18}
\]
Finding a common denominator (which is 180):
\[
\frac{1}{90} = \frac{2}{180}, \quad \frac{1}{60} = \frac{3}{180}, \quad \frac{1}{18} = \frac{10}{180}
\]
Thus,
\[
\text{Combined work rate of A, B, and C} = \frac{2}{180} + \frac{3}{180} + \frac{10}{180} = \frac{15}{180} = \frac{1}{12} \text{ of work per day}
\]
### Step 9: Calculate the time taken by A, B, and C to complete the work together.
If A, B, and C work together, they will complete the work in:
\[
\text{Time} = \frac{1 \text{ unit of work}}{\frac{1}{12} \text{ of work per day}} = 12 \text{ days}
\]
### Final Answer:
A, B, and C together will complete the work in **12 days**.
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