Working 2 hours in a day, P can complete a work in 10 days and working 5 hours in a day, Q can complete the same work in 6 days. Working 3 hours in a day, in how many days both P and Q together can complete the same work?

Options

A. 3 days
B. 6 days
C. 4 days
D. 5 days

Answer

To solve the problem step by step, we need to determine how many days P and Q can complete the work together when they work 3 hours a day.

### Step 1: Calculate the total work done by P
P works for 2 hours a day and completes the work in 10 days.
- Total hours worked by P = 2 hours/day × 10 days = 20 hours.

### Step 2: Calculate the total work done by Q
Q works for 5 hours a day and completes the work in 6 days.
- Total hours worked by Q = 5 hours/day × 6 days = 30 hours.

### Step 3: Find the total work in terms of hours
To find a common measure of work, we can take the LCM of the total hours worked by P and Q.
- LCM of 20 and 30 = 60 hours.

### Step 4: Calculate the work done by P in one day
To find out how much work P does in one day:
- Work done by P in one day = Total work / Total days = 60 hours / 20 hours = 3 units.

### Step 5: Calculate the work done by Q in one day
To find out how much work Q does in one day:
- Work done by Q in one day = Total work / Total days = 60 hours / 30 hours = 2 units.

### Step 6: Calculate the combined work done by P and Q in one day
When P and Q work together:
- Combined work done in one day = Work done by P + Work done by Q = 3 units + 2 units = 5 units.

### Step 7: Calculate how many days it takes for P and Q to complete the work together
Now, we need to find out how many days it will take for them to complete the total work (60 units) when they work together for 3 hours a day.
- Total work = 60 units.
- Work done in one day (when working together) = 5 units.
- Days required = Total work / Combined work per day = 60 units / 5 units = 12 days.

### Step 8: Calculate the total days when they work 3 hours a day
Since they work 3 hours a day, we need to find out how many days it will take them to complete the work.
- Days required = Total work / (Combined work per day × hours worked per day) = 60 units / (5 units) = 12 days.

Thus, P and Q together can complete the work in **12 days**.