If A,B and C can respectively complete a piece of work in 20,24 and 36 days respectively .how many days will they take to complete the work, if they work together?

Options

A. `8(16)/(43)`
B. `6(1)/(4)`
C. `7(19)/(20)`
D. `9(1)/(4)`

Answer

To solve the problem of how many days A, B, and C will take to complete the work if they work together, we can follow these steps:

### Step 1: Determine the work done by each person in one day.

- A can complete the work in 20 days. Therefore, the work done by A in one day is:
\[ \text{Work done by A in one day} = \frac{1}{20} \text{ of the work} \]

- B can complete the work in 24 days. Therefore, the work done by B in one day is:
\[ \text{Work done by B in one day} = \frac{1}{24} \text{ of the work} \]

- C can complete the work in 36 days. Therefore, the work done by C in one day is:
\[ \text{Work done by C in one day} = \frac{1}{36} \text{ of the work} \]

### Step 2: Find 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 the individual contributions:
\[ \text{Total work in one day} = \frac{1}{20} + \frac{1}{24} + \frac{1}{36} \]

### Step 3: Find a common denominator to add the fractions.

The least common multiple (LCM) of 20, 24, and 36 is 720. We convert each fraction:
\[ \frac{1}{20} = \frac{36}{720}, \quad \frac{1}{24} = \frac{30}{720}, \quad \frac{1}{36} = \frac{20}{720} \]

Now, we can add the fractions:
\[ \text{Total work in one day} = \frac{36}{720} + \frac{30}{720} + \frac{20}{720} = \frac{86}{720} \]

### Step 4: Simplify the total work done in one day.

Now, simplify \(\frac{86}{720}\):
\[ \frac{86}{720} = \frac{43}{360} \]

### Step 5: Calculate the number of days to complete the work together.

If A, B, and C together complete \(\frac{43}{360}\) of the work in one day, the total number of days to complete the entire work is:
\[ \text{Total days} = \frac{1}{\frac{43}{360}} = \frac{360}{43} \approx 8.37 \text{ days} \]

Thus, A, B, and C will take approximately 8.37 days to complete the work together.

### Final Answer:
The answer is approximately 8.37 days.

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