What is the product? Use a representation to solve. $$ \begin{array}{ll} ext { 1. } \frac{3}{4} imes \frac{1}{2}= & ext { 2. } \frac{2}{3} imes \frac{5}{6}=\end{array} $$

Question

What is the product? Use a representation to solve. $$ \begin{array}{ll}	ext { 1. } \frac{3}{4} 	imes \frac{1}{2}= & 	ext { 2. } \frac{2}{3} 	imes \frac{5}{6}=\end{array} $$

UpStudy AI · Accepted Answer

Sure! Let's find the products of the given fractions using visual representations called **area models**. An area model helps visualize the multiplication of fractions by representing each fraction as a part of a whole.

---

### Problem 1: $$ \frac{3}{4} \times \frac{1}{2} $$

**Step 1: Draw a Rectangle to Represent 1 Whole**
```
+---------------------+
|                     |
|                     |
|                     |
|                     |
+---------------------+
```

**Step 2: Divide the Rectangle into 4 Equal Parts Horizontally (Representing $$ \frac{3}{4} $$)**

Each part represents $$ \frac{1}{4} $$ of the whole.
```
+----------+----------+----------+----------+
|          |          |          |          |
|          |          |          |          |
|          |          |          |          |
|          |          |          |          |
+----------+----------+----------+----------+
```
- Shade **3 out of 4 parts** to represent $$ \frac{3}{4} $$:
```
+----------+----------+----------+----------+
|██████████|██████████|██████████|          |
|██████████|██████████|██████████|          |
|██████████|██████████|██████████|          |
|██████████|██████████|██████████|          |
+----------+----------+----------+----------+
```

**Step 3: Divide the Rectangle into 2 Equal Parts Vertically (Representing $$ \frac{1}{2} $$)**
```
+----------+----------+----------+----------+
|██████████|██████████|██████████|          |
|██████████|██████████|██████████|          |
|----------+----------+----------+----------+
|██████████|██████████|██████████|          |
|██████████|██████████|██████████|          |
+----------+----------+----------+----------+
```

**Step 4: Identify the Overlapping Shaded Area**
- The overlapping area between the $$ \frac{3}{4} $$ (horizontal shading) and $$ \frac{1}{2} $$ (vertical shading) represents the product.

```
Shaded area (both horizontal and vertical):
+----------+----------+
|██████████|██████████|
|██████████|██████████|
+----------+----------+
```

**Calculation:**
$$
\frac{3}{4} \times \frac{1}{2} = \frac{3 \times 1}{4 \times 2} = \frac{3}{8}
$$

**Answer:** $$ \frac{3}{8} $$

---

### Problem 2: $$ \frac{2}{3} \times \frac{5}{6} $$

**Step 1: Draw a Rectangle to Represent 1 Whole**
```
+---------------------+
|                     |
|                     |
|                     |
|                     |
+---------------------+
```

**Step 2: Divide the Rectangle into 3 Equal Parts Horizontally (Representing $$ \frac{2}{3} $$)**

Each part represents $$ \frac{1}{3} $$ of the whole.
```
+----------+----------+----------+
|          |          |          |
|          |          |          |
|          |          |          |
|          |          |          |
+----------+----------+----------+
```
- Shade **2 out of 3 parts** to represent $$ \frac{2}{3} $$:
```
+----------+----------+----------+
|██████████|██████████|          |
|██████████|██████████|          |
|██████████|██████████|          |
|██████████|██████████|          |
+----------+----------+----------+
```

**Step 3: Divide the Rectangle into 6 Equal Parts Vertically (Representing $$ \frac{5}{6} $$)**
```
+----+----+----+----+----+----+
|████|████|████|    |    |    |
|████|████|████|    |    |    |
|████|████|████|    |    |    |
|████|████|████|    |    |    |
+----+----+----+----+----+----+
```
- Shade **5 out of 6 parts** to represent $$ \frac{5}{6} $$:
```
+----+----+----+----+----+----+
|████|████|████|████|████|    |
|████|████|████|████|████|    |
|████|████|████|████|████|    |
|████|████|████|████|████|    |
+----+----+----+----+----+----+
```

**Step 4: Identify the Overlapping Shaded Area**
- The overlapping area between the $$ \frac{2}{3} $$ (horizontal shading) and $$ \frac{5}{6} $$ (vertical shading) represents the product.

```
Shaded overlapping area:
+----+----+----+----+----+----+
|████|████|████|████|████|    |
|████|████|████|████|████|    |
+----+----+----+----+----+----+
```

Since $$ \frac{2}{3} $$ of the whole is shaded horizontally and $$ \frac{5}{6} $$ is shaded vertically, the overlapping area is $$ \frac{2}{3} \times \frac{5}{6} $$.

**Calculation:**
$$
\frac{2}{3} \times \frac{5}{6} = \frac{2 \times 5}{3 \times 6} = \frac{10}{18}
$$
Simplify the fraction by dividing numerator and denominator by 2:
$$
\frac{10 \div 2}{18 \div 2} = \frac{5}{9}
$$

**Answer:** $$ \frac{5}{9} $$

---

**Summary of Products:**
1. $$ \frac{3}{4} \times \frac{1}{2} = \frac{3}{8} $$
2. $$ \frac{2}{3} \times \frac{5}{6} = \frac{5}{9} $$
The products are: 1. $$ \frac{3}{4} \times \frac{1}{2} = \frac{3}{8} $$ 2. $$ \frac{2}{3} \times \frac{5}{6} = \frac{5}{9} $$

What is the product? Use a representation to solve. \( \begin{array}{ll}\text { 1. } \frac{3}{4} \times \frac{1}{2}= & \text { 2. } \frac{2}{3} \times \frac{5}{6}=\end{array} \)

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