P can complete a work in 6 days working 16 hours a day. Q can complete the same work in 4 days, working 20 hours a day. If both P and Q work together, working 8 hours a day, in how many days can they complete the work?

To solve this problem, we'll determine how much of the work P and Q can complete together in one day and then calculate how many days it will take for them to finish the entire work.

### Step 1: Calculate the total work done by P and Q individually
- **P's work rate:**
  - P can complete the work in 6 days working 16 hours per day.
  - Total hours P works = \(6 \times 16 = 96\) hours.
  - So, P's work rate per hour = \(\frac{1}{96}\) of the work per hour.

- **Q's work rate:**
  - Q can complete the work in 4 days working 20 hours per day.
  - Total hours Q works = \(4 \times 20 = 80\) hours.
  - So, Q's work rate per hour = \(\frac{1}{80}\) of the work per hour.

### Step 2: Calculate the combined work rate when P and Q work together
- Combined work rate per hour = \( \frac{1}{96} + \frac{1}{80} \).

To add these fractions, find a common denominator:

\[
\frac{1}{96} = \frac{5}{480}, \quad \frac{1}{80} = \frac{6}{480}
\]

\[
\text{Combined work rate per hour} = \frac{5}{480} + \frac{6}{480} = \frac{11}{480}
\]

This means that together, P and Q complete \(\frac{11}{480}\) of the work in one hour.

### Step 3: Determine how much work they complete in 8 hours per day
- Work done by P and Q together in 8 hours = \(8 \times \frac{11}{480} = \frac{88}{480} = \frac{11}{60}\) of the work per day.

### Step 4: Calculate the number of days required to complete the work
- Let the number of days required be \(x\).

Since they complete \(\frac{11}{60}\) of the work in one day:

\[
x \times \frac{11}{60} = 1 \quad \Rightarrow \quad x = \frac{60}{11} \approx 5.45 \text{ days}
\]

So, it will take approximately **5.45 days** for P and Q to complete the work when working together for 8 hours a day.

Would you like more details on any step or have any questions?

Here are some related problems you might find interesting:
1. What if P and Q work 10 hours a day together instead of 8 hours?
2. How much time will it take for P and Q to complete the work if P works alone for the first day and then both work together?
3. How many hours a day should P and Q work together to complete the work in exactly 5 days?
4. If P works 6 hours a day and Q works 10 hours a day, how many days will they need together?
5. What if a third worker R can complete the work in 8 days working 12 hours a day? How will that change the total time needed?

**Tip:** When solving work rate problems, always break down the problem into manageable steps and use the concept of "work done per hour" to find the combined work rate.

Discover how to calculate the time required for P and Q to complete a work task together, based on their individual work rates. This problem involves rates of work and time calculations, suitable for advanced high school students.

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