Homework worksheet focusing on complex fractions and unit rates, featuring problems that require simplifying fractions and calculating rates in various real-life scenarios.

Let’s solve each problem step by step, showing all work.

---

1. Simplify: \(\frac{18}{\frac{3}{4}}\)

This is a complex fraction — a fraction divided by another fraction.

To divide by a fraction, multiply by its reciprocal:

\[
\frac{18}{\frac{3}{4}} = 18 \times \frac{4}{3}
\]

Now multiply:

\[
18 \times \frac{4}{3} = \frac{18 \times 4}{3} = \frac{72}{3} = 24
\]

✔ Answer: 24

---

2. Simplify: \(\frac{\frac{3}{6}}{4}\)

First, simplify the numerator: \(\frac{3}{6} = \frac{1}{2}\)

So now we have:

\[
\frac{\frac{1}{2}}{4} = \frac{1}{2} \div 4 = \frac{1}{2} \times \frac{1}{4} = \frac{1}{8}
\]

✔ Answer: \(\frac{1}{8}\)

---

3. Simplify: \(\frac{\frac{1}{3}}{\frac{1}{4}}\)

Divide fractions by multiplying by the reciprocal of the denominator:

\[
\frac{1}{3} \div \frac{1}{4} = \frac{1}{3} \times \frac{4}{1} = \frac{4}{3}
\]

You can also write this as a mixed number: \(1 \frac{1}{3}\), but unless specified, improper fraction is fine.

✔ Answer: \(\frac{4}{3}\)

---

4. Pep Club makes 490 spirit buttons in \(3 \frac{1}{2}\) hours. Find buttons per hour.

We need to find the unit rate: buttons ÷ hours.

First, convert \(3 \frac{1}{2}\) to an improper fraction:

\[
3 \frac{1}{2} = \frac{7}{2}
\]

Now divide:

\[
\frac{490}{\frac{7}{2}} = 490 \times \frac{2}{7}
\]

Simplify:

\[
\frac{490 \times 2}{7} = \frac{980}{7} = 140
\]

✔ Answer: 140 buttons per hour

---

5. Mary bought \(2 \frac{1}{2}\) yards of fabric for $15. What was cost per yard?

Unit rate: total cost ÷ total yards.

Convert \(2 \frac{1}{2}\) to improper fraction:

\[
2 \frac{1}{2} = \frac{5}{2}
\]

Now divide:

\[
\frac{15}{\frac{5}{2}} = 15 \times \frac{2}{5} = \frac{30}{5} = 6
\]

✔ Answer: $6 per yard

---

6. Doug rowed \(3 \frac{1}{2}\) miles in \(\frac{1}{2}\) hour. Average speed in miles per hour?

Speed = distance ÷ time

Convert \(3 \frac{1}{2}\) to improper fraction:

\[
3 \frac{1}{2} = \frac{7}{2}
\]

Time = \(\frac{1}{2}\) hour

So,

\[
\text{Speed} = \frac{\frac{7}{2}}{\frac{1}{2}} = \frac{7}{2} \times \frac{2}{1} = \frac{14}{2} = 7
\]

✔ Answer: 7 miles per hour

---

7. Monica reads \(7 \frac{1}{2}\) pages in 9 minutes. Average reading rate in pages per minute?

Rate = pages ÷ minutes

Convert \(7 \frac{1}{2}\) to improper fraction:

\[
7 \frac{1}{2} = \frac{15}{2}
\]

Now divide:

\[
\frac{\frac{15}{2}}{9} = \frac{15}{2} \times \frac{1}{9} = \frac{15}{18}
\]

Simplify \(\frac{15}{18}\) by dividing numerator and denominator by 3:

\[
\frac{15 \div 3}{18 \div 3} = \frac{5}{6}
\]

✔ Answer: \(\frac{5}{6}\) pages per minute

---

## ✔ Final Answers Summary:

1. 24
2. \(\frac{1}{8}\)
3. \(\frac{4}{3}\)
4. 140 buttons per hour
5. $6 per yard
6. 7 miles per hour
7. \(\frac{5}{6}\) pages per minute

Let me know if you’d like any step explained further!