To find the total time Josh spent doing these activities, we will add up all the given hours:

• School time: $6 \frac{1}{2}$ hours
• Work time: $3 \frac{2}{3}$ hours
• Homework time: $2 \frac{3}{4}$ hours
• Television time: $1 \frac{1}{3}$ hours

Step 1: Convert Mixed Numbers to Improper Fractions

1. For $6 \frac{1}{2}$: $6 \frac{1}{2} = \frac{6 \times 2 + 1}{2} = \frac{13}{2}$

2. For $3 \frac{2}{3}$: $3 \frac{2}{3} = \frac{3 \times 3 + 2}{3} = \frac{11}{3}$

3. For $2 \frac{3}{4}$: $2 \frac{3}{4} = \frac{2 \times 4 + 3}{4} = \frac{11}{4}$

4. For $1 \frac{1}{3}$: $1 \frac{1}{3} = \frac{1 \times 3 + 1}{3} = \frac{4}{3}$

Step 2: Find a Common Denominator

The denominators are 2, 3, and 4. The least common multiple (LCM) of 2, 3, and 4 is 12.

Step 3: Convert Fractions to Have a Common Denominator

1. Convert $\frac{13}{2}$: $\frac{13}{2} = \frac{13 \times 6}{12} = \frac{78}{12}$

2. Convert $\frac{11}{3}$: $\frac{11}{3} = \frac{11 \times 4}{12} = \frac{44}{12}$

3. Convert $\frac{11}{4}$: $\frac{11}{4} = \frac{11 \times 3}{12} = \frac{33}{12}$

4. Convert $\frac{4}{3}$: $\frac{4}{3} = \frac{4 \times 4}{12} = \frac{16}{12}$

Step 4: Add All the Fractions

$\frac{78}{12} + \frac{44}{12} + \frac{33}{12} + \frac{16}{12} = \frac{171}{12}$

Step 5: Simplify the Fraction

$\frac{171}{12} \approx 14 \frac{3}{12}$

We can simplify $\frac{3}{12}$ to $\frac{1}{4}$. Thus, the final answer is:

$14 \frac{1}{4} \text{ hours}$

Answer:

Josh spent a total of 14 1/4 hours doing these activities.

Would you like a breakdown of any step or further details on how the calculations were made?

Here are 5 relative questions to expand:

1. How can we find the least common multiple (LCM) for more complex denominators?
2. What’s the quickest way to add multiple fractions with different denominators?
3. How do you convert a mixed number into an improper fraction?
4. Can you explain why it is necessary to simplify fractions?
5. What strategies can you use when comparing mixed numbers directly?

Tip: When adding mixed numbers, convert them to improper fractions first to avoid mistakes with addition and denominators.