Algebra 1 Practice - Solving a Proportion (Example 2)

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Solving a proportion involves finding the value of the variable that makes two ratios equal. A proportion is an equation of the form \(\frac{a}{b} = \frac{c}{d}\), where \(a\), \(b\), \(c\), and \(d\) are numbers and \(b\) and \(d\) are not zero. Here’s a detailed guide on solving proportions:
General Steps:
1. **Set Up the Proportion**: Write the equation in the form \(\frac{a}{b} = \frac{c}{d}\).
2. **Cross Multiply**: Multiply the numerator of one ratio by the denominator of the other ratio.
3. **Solve the Resulting Equation**: Solve the resulting linear equation for the variable.
4. **Check Your Solution**: Substitute your solution back into the original proportion to ensure both sides are equal.
Example 1: Simple Proportion
**Proportion**: \(\frac{x}{4} = \frac{3}{8}\)
1. **Set Up the Proportion**:
\[
\frac{x}{4} = \frac{3}{8}
\]
2. **Cross Multiply**:
\[
x \cdot 8 = 4 \cdot 3
\]
Simplifies to:
\[
8x = 12
\]
3. **Solve the Resulting Equation**:
\[
x = \frac{12}{8}
\]
Simplifies to:
\[
x = \frac{3}{2} \quad \text{or } 1.5
\]
4. **Check Your Solution**:
Substitute \(x = 1.5\) back into the original proportion:
\[
\frac{1.5}{4} = \frac{3}{8}
\]
Simplifies to:
\[
0.375 = 0.375 \quad \text{(True, so the solution is correct)}
\]
Example 2: Proportion with a Variable in the Denominator
**Proportion**: \(\frac{5}{x} = \frac{10}{6}\)
1. **Set Up the Proportion**:
\[
\frac{5}{x} = \frac{10}{6}
\]
2. **Cross Multiply**:
\[
5 \cdot 6 = 10 \cdot x
\]
Simplifies to:
\[
30 = 10x
\]
3. **Solve the Resulting Equation**:
\[
x = \frac{30}{10}
\]
Simplifies to:
\[
x = 3
\]
4. **Check Your Solution**:
Substitute \(x = 3\) back into the original proportion:
\[
\frac{5}{3} = \frac{10}{6}
\]
Simplifies to:
\[
\frac{5}{3} = \frac{5}{3} \quad \text{(True, so the solution is correct)}
\]
Example 3: Proportion Involving Variables on Both Sides
**Proportion**: \(\frac{2x + 1}{5} = \frac{3x - 2}{4}\)
1. **Set Up the Proportion**:
\[
\frac{2x + 1}{5} = \frac{3x - 2}{4}
\]
2. **Cross Multiply**:
\[
(2x + 1) \cdot 4 = (3x - 2) \cdot 5
\]
Simplifies to:
\[
8x + 4 = 15x - 10
\]
3. **Solve the Resulting Equation**:
- Subtract \(8x\) from both sides:
\[
4 = 7x - 10
\]
- Add 10 to both sides:
\[
14 = 7x
\]
- Divide both sides by 7:
\[
x = 2
\]
4. **Check Your Solution**:
Substitute \(x = 2\) back into the original proportion:
\[
\frac{2(2) + 1}{5} = \frac{3(2) - 2}{4}
\]
Simplifies to:
\[
\frac{4 + 1}{5} = \frac{6 - 2}{4}
\]
\[
\frac{5}{5} = \frac{4}{4}
\]
\[
1 = 1 \quad \text{(True, so the solution is correct)}
\]
Key Points:
- **Cross Multiplication**: Multiply diagonally across the equal sign to create a linear equation.
- **Simplify and Solve**: After cross-multiplying, solve the resulting equation as you would any linear equation.
- **Verification**: Always check your solution by substituting it back into the original proportion to ensure both sides are equal.
By following these steps, you can solve proportions accurately and systematically.
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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa
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