Multiplying Fractions Worksheets with Answer Key - Free Printable

To solve the problem of multiplying fractions using the area model, we need to follow these steps:

1. Understand the Area Model: The area model represents a rectangle divided into smaller sections. Each fraction corresponds to a part of the rectangle.
2. Identify the Fractions: Determine which parts of the rectangle are shaded for each fraction.
3. Find the Overlapping Area: The product of the two fractions is represented by the overlapping shaded area in the rectangle.
4. Count the Overlapping Squares: Count the number of squares in the overlapping shaded area and compare it to the total number of squares in the rectangle.

Let's solve each problem step by step:

---

Problem ②:

\[
\frac{2}{5} \times \frac{3}{4}
\]

- The rectangle is divided into 5 columns and 4 rows, making a total of \(5 \times 4 = 20\) squares.
- \(\frac{2}{5}\) means 2 out of 5 columns are shaded.
- \(\frac{3}{4}\) means 3 out of 4 rows are shaded.
- The overlapping area (shaded in both directions) consists of \(2 \times 3 = 6\) squares.
- Therefore, the product is \(\frac{6}{20}\), which simplifies to \(\frac{3}{10}\).

\[
\boxed{\frac{3}{10}}
\]

---

Problem ③:

\[
\frac{2}{5} \times \frac{5}{8}
\]

- The rectangle is divided into 5 columns and 8 rows, making a total of \(5 \times 8 = 40\) squares.
- \(\frac{2}{5}\) means 2 out of 5 columns are shaded.
- \(\frac{5}{8}\) means 5 out of 8 rows are shaded.
- The overlapping area consists of \(2 \times 5 = 10\) squares.
- Therefore, the product is \(\frac{10}{40}\), which simplifies to \(\frac{1}{4}\).

\[
\boxed{\frac{1}{4}}
\]

---

Problem ④:

\[
\frac{4}{12} \times \frac{2}{3}
\]

- The rectangle is divided into 12 columns and 3 rows, making a total of \(12 \times 3 = 36\) squares.
- \(\frac{4}{12}\) means 4 out of 12 columns are shaded.
- \(\frac{2}{3}\) means 2 out of 3 rows are shaded.
- The overlapping area consists of \(4 \times 2 = 8\) squares.
- Therefore, the product is \(\frac{8}{36}\), which simplifies to \(\frac{2}{9}\).

\[
\boxed{\frac{2}{9}}
\]

---

Problem ⑤:

\[
\frac{3}{6} \times \frac{2}{4}
\]

- The rectangle is divided into 6 columns and 4 rows, making a total of \(6 \times 4 = 24\) squares.
- \(\frac{3}{6}\) means 3 out of 6 columns are shaded.
- \(\frac{2}{4}\) means 2 out of 4 rows are shaded.
- The overlapping area consists of \(3 \times 2 = 6\) squares.
- Therefore, the product is \(\frac{6}{24}\), which simplifies to \(\frac{1}{4}\).

\[
\boxed{\frac{1}{4}}
\]

---

Problem ⑥:

\[
\frac{5}{6} \times \frac{5}{8}
\]

- The rectangle is divided into 6 columns and 8 rows, making a total of \(6 \times 8 = 48\) squares.
- \(\frac{5}{6}\) means 5 out of 6 columns are shaded.
- \(\frac{5}{8}\) means 5 out of 8 rows are shaded.
- The overlapping area consists of \(5 \times 5 = 25\) squares.
- Therefore, the product is \(\frac{25}{48}\).

\[
\boxed{\frac{25}{48}}
\]

---

Problem ⑦:

\[
\frac{2}{7} \times \frac{2}{5}
\]

- The rectangle is divided into 7 columns and 5 rows, making a total of \(7 \times 5 = 35\) squares.
- \(\frac{2}{7}\) means 2 out of 7 columns are shaded.
- \(\frac{2}{5}\) means 2 out of 5 rows are shaded.
- The overlapping area consists of \(2 \times 2 = 4\) squares.
- Therefore, the product is \(\frac{4}{35}\).

\[
\boxed{\frac{4}{35}}
\]

---

Problem ⑧:

\[
\frac{5}{9} \times \frac{3}{5}
\]

- The rectangle is divided into 9 columns and 5 rows, making a total of \(9 \times 5 = 45\) squares.
- \(\frac{5}{9}\) means 5 out of 9 columns are shaded.
- \(\frac{3}{5}\) means 3 out of 5 rows are shaded.
- The overlapping area consists of \(5 \times 3 = 15\) squares.
- Therefore, the product is \(\frac{15}{45}\), which simplifies to \(\frac{1}{3}\).

\[
\boxed{\frac{1}{3}}
\]

---

Problem ⑨:

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

- The rectangle is divided into 2 columns and 7 rows, making a total of \(2 \times 7 = 14\) squares.
- \(\frac{1}{2}\) means 1 out of 2 columns are shaded.
- \(\frac{3}{7}\) means 3 out of 7 rows are shaded.
- The overlapping area consists of \(1 \times 3 = 3\) squares.
- Therefore, the product is \(\frac{3}{14}\).

\[
\boxed{\frac{3}{14}}
\]

---

Problem ⑩:

\[
\frac{7}{8} \times \frac{1}{2}
\]

- The rectangle is divided into 8 columns and 2 rows, making a total of \(8 \times 2 = 16\) squares.
- \(\frac{7}{8}\) means 7 out of 8 columns are shaded.
- \(\frac{1}{2}\) means 1 out of 2 rows are shaded.
- The overlapping area consists of \(7 \times 1 = 7\) squares.
- Therefore, the product is \(\frac{7}{16}\).

\[
\boxed{\frac{7}{16}}
\]

---

Problem ⑪:

\[
\frac{4}{5} \times \frac{7}{11}
\]

- The rectangle is divided into 5 columns and 11 rows, making a total of \(5 \times 11 = 55\) squares.
- \(\frac{4}{5}\) means 4 out of 5 columns are shaded.
- \(\frac{7}{11}\) means 7 out of 11 rows are shaded.
- The overlapping area consists of \(4 \times 7 = 28\) squares.
- Therefore, the product is \(\frac{28}{55}\).

\[
\boxed{\frac{28}{55}}
\]

---

Problem ⑫:

\[
\frac{3}{4} \times \frac{3}{5}
\]

- The rectangle is divided into 4 columns and 5 rows, making a total of \(4 \times 5 = 20\) squares.
- \(\frac{3}{4}\) means 3 out of 4 columns are shaded.
- \(\frac{3}{5}\) means 3 out of 5 rows are shaded.
- The overlapping area consists of \(3 \times 3 = 9\) squares.
- Therefore, the product is \(\frac{9}{20}\).

\[
\boxed{\frac{9}{20}}
\]

---

Final Answers:

\[
\boxed{
\begin{array}{ccc}
\text{②} & \frac{3}{10} & \\
\text{③} & \frac{1}{4} & \\
\text{④} & \frac{2}{9} & \\
\text{⑤} & \frac{1}{4} & \\
\text{⑥} & \frac{25}{48} & \\
\text{⑦} & \frac{4}{35} & \\
\text{⑧} & \frac{1}{3} & \\
\text{⑨} & \frac{3}{14} & \\
\text{⑩} & \frac{7}{16} & \\
\text{⑪} & \frac{28}{55} & \\
\text{⑫} & \frac{9}{20} &
\end{array}
}
\]