The question from the image is as follows:

Question 5:
2 people working at the same rate will take 6 hours to paint a room. Assuming that they all work at this rate, how long will it take 3 people to paint the room?

Solution:

1. Rate of work for 2 people: If 2 people take 6 hours to paint the room, the total amount of work done is 1 room. Hence, the rate at which 2 people work together is: $\text{Rate of 2 people} = \frac{1 \text{ room}}{6 \text{ hours}} = \frac{1}{6} \text{ room per hour}.$

2. Rate of work for 1 person: Since the rate of work is directly proportional to the number of people, the rate for 1 person would be half the rate of 2 people: $\text{Rate of 1 person} = \frac{1}{2} \times \frac{1}{6} = \frac{1}{12} \text{ room per hour}.$

3. Rate of work for 3 people: The rate of work for 3 people would be three times the rate of 1 person: $\text{Rate of 3 people} = 3 \times \frac{1}{12} = \frac{3}{12} = \frac{1}{4} \text{ room per hour}.$

4. Time taken for 3 people: The time taken to paint 1 room at a rate of $\frac{1}{4}$ room per hour is: $\text{Time} = \frac{1 \text{ room}}{\frac{1}{4} \text{ room per hour}} = 4 \text{ hours}.$

Thus, it will take 3 people 4 hours to paint the room.

Would you like further details on any step? Here are 5 related questions to deepen your understanding:

1. What if 4 people worked together instead of 3—how long would it take them?
2. How would the time change if the number of people painting doubled?
3. If each person could paint only half as fast, how long would it take 3 people to paint the room?
4. What if 3 people started working, but one person left after 2 hours—how long would it take to finish the room?
5. How much of the room would 2 people complete in 3 hours?

Tip: For problems involving rates, always remember that the total work done (like painting a room) is the product of the rate of work and the time spent working.