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. `9 1/4`
D. `7 (19)/(20)`

Answer

To solve the problem of how many days A, B, and C will take to complete the 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, so in one day, A does:
\[ \text{Work done by A in one day} = \frac{1}{20} \]

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

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

### 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 in one day, we add their individual contributions:
\[ \text{Total work in one day} = \frac{1}{20} + \frac{1}{24} + \frac{1}{36} \]

### Step 3: Calculate the least common multiple (LCM) of the denominators.
The LCM of 20, 24, and 36 is 360. We will convert each fraction to have a common denominator of 360:
- For A:
\[ \frac{1}{20} = \frac{18}{360} \]
- For B:
\[ \frac{1}{24} = \frac{15}{360} \]
- For C:
\[ \frac{1}{36} = \frac{10}{360} \]

### Step 4: Add the fractions.
Now we can add the fractions:
\[ \text{Total work in one day} = \frac{18}{360} + \frac{15}{360} + \frac{10}{360} = \frac{43}{360} \]

### Step 5: Calculate the total time taken to complete the work.
If A, B, and C together can complete \(\frac{43}{360}\) of the work in one day, we need to find out how many days it will take to complete the entire work (1 unit of work):
\[ \text{Time taken} = \frac{1 \text{ unit of work}}{\frac{43}{360}} = \frac{360}{43} \]

### Step 6: Simplify the result.
Calculating \(\frac{360}{43}\) gives approximately:
\[ \frac{360}{43} \approx 8.37 \text{ days} \]
This can also be expressed as \(8\) days and \(\frac{16}{43}\) of a day.

### Final Answer:
A, B, and C together will take approximately \(8\) days and \(\frac{16}{43}\) of a day to complete the work.

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