The problem asks to find the total time spent on homework over Tuesday and Wednesday. Let's break it down:

• On Tuesday, you spend $2 \frac{1}{2}$ hours on homework.
• On Wednesday, you spend $1 \frac{2}{3}$ hours on homework.

We need to add these two mixed numbers together.

Step 1: Convert the mixed numbers to improper fractions.

• $2 \frac{1}{2} = \frac{5}{2}$
• $1 \frac{2}{3} = \frac{5}{3}$

Step 2: Find a common denominator for the fractions.

The denominators are 2 and 3. The least common denominator (LCD) is 6.

• Convert $\frac{5}{2}$ to an equivalent fraction with a denominator of 6: $\frac{5}{2} = \frac{15}{6}$

• Convert $\frac{5}{3}$ to an equivalent fraction with a denominator of 6: $\frac{5}{3} = \frac{10}{6}$

Step 3: Add the two fractions.

$\frac{15}{6} + \frac{10}{6} = \frac{25}{6}$

Step 4: Convert the improper fraction back to a mixed number.

$\frac{25}{6} = 4 \frac{1}{6}$

Conclusion:

The total time spent on homework on Tuesday and Wednesday is 4 hours and 10 minutes (since $\frac{1}{6}$ of an hour is 10 minutes).

Would you like more details on any step? Here are some related questions to consider:

1. How do you find the least common denominator between two fractions?
2. How do you convert improper fractions back to mixed numbers?
3. What is the process to add mixed numbers without converting to improper fractions?
4. How would this process change if the fractions had larger denominators?
5. What are some other methods to solve similar problems involving time?

Tip: When working with fractions, always check if you can simplify before converting to mixed numbers!