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Let's break down the problem step by step:

- **Haley's rate**: Haley can clean the room in 2 hours, so her rate of cleaning is \( \frac{1}{2} \) of the room per hour.
- **Alex's rate**: Alex can clean the room in 3 hours, so his rate is \( \frac{1}{3} \) of the room per hour.

### Step 1: Find how much of the room Haley cleans in 1 hour.
Since Haley cleans \( \frac{1}{2} \) of the room in 1 hour, after 1 hour of cleaning, she has completed:
\[
\frac{1}{2} \times 1 = \frac{1}{2} \text{ of the room.}
\]
So, Haley has cleaned half of the room.

### Step 2: Find how much is left for Alex to clean.
The total room is \( 1 \) unit. Haley cleaned \( \frac{1}{2} \), so the remaining part for Alex to clean is:
\[
1 - \frac{1}{2} = \frac{1}{2}.
\]

### Step 3: Find the time it takes Alex to clean the remaining half of the room.
Alex's rate is \( \frac{1}{3} \) of the room per hour. To clean \( \frac{1}{2} \) of the room, the time it will take Alex is:
\[
\text{Time} = \frac{\frac{1}{2}}{\frac{1}{3}} = \frac{1}{2} \times 3 = \frac{3}{2} \text{ hours} = 1.5 \text{ hours}.
\]

### Final Answer:
It took Alex 1.5 hours to clean the remaining part of the room.

Would you like more details or have any other questions?

Here are 5 related questions to explore this concept further:
1. What if Haley cleaned for 30 minutes instead of 1 hour? How long would it take Alex then?
2. If Haley and Alex worked together for the entire time, how long would it take them to finish the room?
3. What would happen if Alex could clean the room in 1.5 hours instead? How would that affect the time needed?
4. How long would it take if Haley cleaned for 1 hour, and they both finished the remaining part together?
5. What would happen if Haley and Alex had to clean a larger space with similar rates? How would the time scale?

**Tip**: Always break down work-rate problems by calculating how much work each person completes per hour or minute.

Haley and Alex are sharing a room. Haley can clean their room in 2 hours, while Alex can clean it in 3 hours. Haley started cleaning their room for 1 hour and Alex continued the rest. How long did it take Alex to clean the room?

Solve this work rate problem involving Haley and Alex cleaning a room. Learn how to calculate how long it took Alex to finish cleaning after Haley did her part. Suitable for middle school students in Grades 6-8.

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