P, Q and R alone can complete a work in 12. 15 and 20 days respectively. In how many days will they together complete the same work?

Options

A. A) 4
B. B) 5
C. C) 6
D. D) 3.5

Answer

To solve the problem of how many days P, Q, and R will take together to complete the work, we can follow these steps:

### Step 1: Determine the work done by each person in one day.
- **P** can complete the work in 12 days. Therefore, the work done by P in one day is:
\[ \text{Work done by P in one day} = \frac{1}{12} \text{ of the work} \]

- **Q** can complete the work in 15 days. Therefore, the work done by Q in one day is:
\[ \text{Work done by Q in one day} = \frac{1}{15} \text{ of the work} \]

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

### Step 2: Calculate the total work done by P, Q, and R in one day.
Now, we can add the work done by each person in one day:
\[ \text{Total work done in one day} = \frac{1}{12} + \frac{1}{15} + \frac{1}{20} \]

### Step 3: Find a common denominator to add the fractions.
The least common multiple (LCM) of 12, 15, and 20 is 60. We will convert each fraction:
- \(\frac{1}{12} = \frac{5}{60}\)
- \(\frac{1}{15} = \frac{4}{60}\)
- \(\frac{1}{20} = \frac{3}{60}\)

Now, we can add them:
\[ \text{Total work done in one day} = \frac{5}{60} + \frac{4}{60} + \frac{3}{60} = \frac{12}{60} = \frac{1}{5} \text{ of the work} \]

### Step 4: Calculate the total time taken to complete the work together.
If together they complete \(\frac{1}{5}\) of the work in one day, then the total time taken to complete the entire work is:
\[ \text{Total time} = \frac{1}{\frac{1}{5}} = 5 \text{ days} \]

### Final Answer:
P, Q, and R together will complete the work in **5 days**.

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