Worksheet for solving proportions with examples and answers.

It looks like you've uploaded a page from a math textbook or worksheet titled "Solving Proportions", which explains how to solve proportions using the cross-multiplication method. The page includes:

- An explanation of what a proportion is.
- Example 1 and Example 2 with step-by-step solutions.
- A set of practice problems (numbered 1 through 8).
- Answers at the bottom.

Let’s go through the content and solve each problem step by step, explaining the reasoning.

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🔷 What Is a Proportion?

A proportion is an equation stating that two ratios are equal. For example:

$$
\frac{a}{b} = \frac{c}{d}
$$

To solve such equations, we use cross multiplication:

$$
a \cdot d = b \cdot c
$$

Then solve for the unknown variable.

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✔ Example 1:

Solve:
$$
\frac{x}{3} = \frac{4}{6}
$$

Step 1: Cross multiply:
$$
x \cdot 6 = 3 \cdot 4
$$
$$
6x = 12
$$

Step 2: Divide both sides by 6:
$$
x = \frac{12}{6} = 2
$$

✔ Answer: $ x = 2 $

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✔ Example 2:

Solve:
$$
\frac{2x - 1}{5} = \frac{x + 3}{4}
$$

Step 1: Cross multiply:
$$
(2x - 1) \cdot 4 = 5 \cdot (x + 3)
$$

Step 2: Expand both sides:
$$
8x - 4 = 5x + 15
$$

Step 3: Subtract $5x$ from both sides:
$$
3x - 4 = 15
$$

Step 4: Add 4 to both sides:
$$
3x = 19
$$

Step 5: Divide by 3:
$$
x = \frac{19}{3}
$$

✔ Answer: $ x = \frac{19}{3} $

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📝 Now Solve the Practice Problems:

#### 1. $ \frac{x}{2} = \frac{3}{4} $

Cross multiply:
$$
4x = 6 \Rightarrow x = \frac{6}{4} = \frac{3}{2}
$$

✔ $ x = \frac{3}{2} $

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#### 2. $ \frac{2}{x} = \frac{6}{9} $

Cross multiply:
$$
2 \cdot 9 = 6x \Rightarrow 18 = 6x \Rightarrow x = 3
$$

✔ $ x = 3 $

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#### 3. $ \frac{5}{x} = \frac{10}{12} $

Cross multiply:
$$
5 \cdot 12 = 10x \Rightarrow 60 = 10x \Rightarrow x = 6
$$

✔ $ x = 6 $

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#### 4. $ \frac{3}{4} = \frac{x}{12} $

Cross multiply:
$$
3 \cdot 12 = 4x \Rightarrow 36 = 4x \Rightarrow x = 9
$$

✔ $ x = 9 $

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#### 5. $ \frac{x}{7} = \frac{5}{14} $

Cross multiply:
$$
14x = 35 \Rightarrow x = \frac{35}{14} = \frac{5}{2}
$$

✔ $ x = \frac{5}{2} $

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#### 6. $ \frac{4}{x} = \frac{2}{3} $

Cross multiply:
$$
4 \cdot 3 = 2x \Rightarrow 12 = 2x \Rightarrow x = 6
$$

✔ $ x = 6 $

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#### 7. $ \frac{3x}{4} = \frac{6}{8} $

Simplify right side: $ \frac{6}{8} = \frac{3}{4} $, so:

$$
\frac{3x}{4} = \frac{3}{4}
$$

Multiply both sides by 4:
$$
3x = 3 \Rightarrow x = 1
$$

✔ $ x = 1 $

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#### 8. $ \frac{x + 2}{3} = \frac{4}{6} $

Simplify right side: $ \frac{4}{6} = \frac{2}{3} $

So:
$$
\frac{x + 2}{3} = \frac{2}{3}
$$

Multiply both sides by 3:
$$
x + 2 = 2 \Rightarrow x = 0
$$

✔ $ x = 0 $

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✔ Final Answers (Matching the Provided Answers):

| Problem | Answer |
|--------|--------|
| 1 | $ \frac{3}{2} $ |
| 2 | 3 |
| 3 | 6 |
| 4 | 9 |
| 5 | $ \frac{5}{2} $ |
| 6 | 6 |
| 7 | 1 |
| 8 | 0 |

These match the answers listed at the bottom of the page.

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🧠 Key Tips for Solving Proportions:

1. Always cross-multiply: If $ \frac{a}{b} = \frac{c}{d} $, then $ a \cdot d = b \cdot c $
2. Simplify fractions first if possible.
3. Distribute carefully when variables are in expressions (like $2x - 1$).
4. Check your answer by plugging it back into the original proportion.

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