If P can do a work in 6 days and Q Can do the same work in
8 days. R can do the same work in 12 days. If they are
woeking together then the work will be completed in how
many days?

Options

A. `7/3 days `
B. ` 8/3 `days
C. `10/3 days`
D. ` 11/3 days `

Explanation

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

### Step 1: Determine the work rates of P, Q, and R
- P can complete the work in 6 days, so P's work rate is \( \frac{1}{6} \) of the work per day.
- Q can complete the work in 8 days, so Q's work rate is \( \frac{1}{8} \) of the work per day.
- R can complete the work in 12 days, so R's work rate is \( \frac{1}{12} \) of the work per day.

### Step 2: Find a common denominator
To add the work rates, we need a common denominator. The least common multiple (LCM) of 6, 8, and 12 is 24.

### Step 3: Convert each work rate to the common denominator
- For P: \( \frac{1}{6} = \frac{4}{24} \)
- For Q: \( \frac{1}{8} = \frac{3}{24} \)
- For R: \( \frac{1}{12} = \frac{2}{24} \)

### Step 4: Add the work rates together
Now, we can add the work rates:
\[ \text{Total work rate} = \frac{4}{24} + \frac{3}{24} + \frac{2}{24} = \frac{9}{24} \]

### Step 5: Simplify the total work rate
The total work rate can be simplified:
\[ \frac{9}{24} = \frac{3}{8} \]
This means together, they can complete \( \frac{3}{8} \) of the work in one day.

### Step 6: Calculate the time taken to complete the work
To find out how many days it will take to complete the entire work, we take the reciprocal of the total work rate:
\[ \text{Time taken} = \frac{1}{\frac{3}{8}} = \frac{8}{3} \text{ days} \]

### Final Answer
Thus, P, Q, and R together will complete the work in \( \frac{8}{3} \) days, which is approximately 2.67 days or 2 days and 8 hours.

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