A can do `1/2` of a piece of work in 5 days, B can do `3/5` of the same work in 9 days and C can do `2/3` work in 8 days. In how many can the three of them together do the work ?

Answer

To solve the problem, we will first determine how much work each person (A, B, and C) can do in one day, and then we will find out how long it will take for all three of them to complete the work together.

### Step 1: Calculate the work done by A in one day.
A can do \( \frac{1}{2} \) of the work in 5 days. Therefore, the work done by A in one day is:
\[ \text{Work done by A in one day} = \frac{1/2}{5} = \frac{1}{10} \]

### Step 2: Calculate the work done by B in one day.
B can do \( \frac{3}{5} \) of the work in 9 days. Therefore, the work done by B in one day is:
\[ \text{Work done by B in one day} = \frac{3/5}{9} = \frac{3}{45} = \frac{1}{15} \]

### Step 3: Calculate the work done by C in one day.
C can do \( \frac{2}{3} \) of the work in 8 days. Therefore, the work done by C in one day is:
\[ \text{Work done by C in one day} = \frac{2/3}{8} = \frac{2}{24} = \frac{1}{12} \]

### Step 4: Calculate the total work done by A, B, and C together in one day.
Now, we will add the work done by A, B, and C in one day:
\[ \text{Total work done in one day} = \frac{1}{10} + \frac{1}{15} + \frac{1}{12} \]

To add these fractions, we need a common denominator. The least common multiple (LCM) of 10, 15, and 12 is 60.

Now, we convert each fraction:
- \( \frac{1}{10} = \frac{6}{60} \)
- \( \frac{1}{15} = \frac{4}{60} \)
- \( \frac{1}{12} = \frac{5}{60} \)

Adding these together:
\[ \text{Total work done in one day} = \frac{6}{60} + \frac{4}{60} + \frac{5}{60} = \frac{15}{60} = \frac{1}{4} \]

### Step 5: Calculate the time taken to complete the work together.
If A, B, and C together can do \( \frac{1}{4} \) of the work in one day, then they can complete the whole work in:
\[ \text{Time taken} = \frac{1}{\text{Total work done in one day}} = \frac{1}{1/4} = 4 \text{ days} \]

### Final Answer:
The three of them together can complete the work in **4 days**.

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