Let's break down and solve the question in the image.

a) How long will it take 3 people to paint the room if all work at the same rate?

We know that:

• 2 people take 9 hours to paint the room.
• Since 3 people are working at the same rate, we can calculate how long it would take by using the idea of "work done."

The amount of work done is proportional to the number of people and the time taken. The work is the same whether 2 or 3 people work on it, so we can set up this proportion:

$\text{Work by 2 people} = \text{Work by 3 people}$ $2 \times 9 = 3 \times T$ Where $T$ is the time taken by 3 people.

Solving for $T$:

$T = \frac{2 \times 9}{3} = 6 \text{ hours}$

So, it will take 6 hours for 3 people to paint the room.

b) If the third person works at a faster rate, how does this affect the time to paint the room?

If the third person works at a faster rate, this means more work is being done in the same amount of time. Therefore, the time taken to paint the room will be shorter (faster).

So the correct answer is A: faster.

Would you like a more detailed explanation or have any questions?

Here are 5 related questions to think about:

1. What would happen if 4 people were painting the room at the same rate?
2. How would you calculate the time if the third person was working at twice the speed of the others?
3. What if one person worked slower than the other two, how would that change the time taken?
4. Can you generalize a formula for any number of people painting a room?
5. How would the total work change if two rooms needed to be painted instead of one?

Tip: When multiple people work together on a task, you can always calculate the time required by considering the combined rate of work they contribute.