Tracy can wallpaper 3 rooms in a new house in 12 hours. Together with her trainee, they can wallpaper the same 3 rooms in 8 hours. The question is: How long would it take the trainee alone to wallpaper the 3 rooms?

Let Tracy's wallpapering rate be \( R_T \) rooms per hour, and the trainee's rate be \( R_{tr} \) rooms per hour.

### Step 1: Calculate Tracy's rate
Tracy can wallpaper 3 rooms in 12 hours. Therefore, her rate is:
\[
R_T = \frac{3}{12} = 0.25 \text{ rooms per hour.}
\]

### Step 2: Combined rate of Tracy and the trainee
Together, Tracy and her trainee can wallpaper the 3 rooms in 8 hours. So their combined rate is:
\[
R_T + R_{tr} = \frac{3}{8} = 0.375 \text{ rooms per hour.}
\]

### Step 3: Solve for the trainee's rate
From the combined rate equation:
\[
R_T + R_{tr} = 0.375
\]
Substitute \( R_T = 0.25 \):
\[
0.25 + R_{tr} = 0.375
\]
\[
R_{tr} = 0.375 - 0.25 = 0.125 \text{ rooms per hour.}
\]

### Step 4: Time for the trainee alone to wallpaper 3 rooms
The time it takes the trainee to wallpaper the 3 rooms is:
\[
\text{Time} = \frac{3 \text{ rooms}}{0.125 \text{ rooms per hour}} = 24 \text{ hours.}
\]

Thus, it would take the trainee **24 hours** to wallpaper the 3 rooms alone.

Do you want further details or explanations? Here are some related questions to explore:

1. If Tracy improves her rate by 20%, how long would it take her to wallpaper the 3 rooms?
2. How long would it take Tracy and two trainees, each working at the same rate, to wallpaper the 3 rooms?
3. If Tracy and her trainee take breaks, reducing their combined efficiency by 10%, how long would they take to complete the 3 rooms?
4. What is the combined rate if the trainee speeds up by 50%?
5. How many rooms can Tracy wallpaper in 24 hours if she works continuously?

**Tip**: When dealing with rates, always express tasks completed per unit of time to simplify calculations.

Find out how long it would take for the trainee to wallpaper 3 rooms alone. This problem explores the concept of work rates, using algebra to solve work problems. We break down the calculations step-by-step, from finding Tracy's rate to solving for the trainee's time. Perfect for students in Grades 6-8.

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