How to solve proportions using cross-multiplication step-by-step

📚 Understanding Proportions and Cross-Multiplication

A proportion is a statement that two ratios are equal. Ratios compare two quantities. Cross-multiplication is a technique used to solve proportions by multiplying the numerator of one ratio by the denominator of the other ratio, and vice versa.

📜 A Brief History

The concept of proportions dates back to ancient civilizations, including the Egyptians and Babylonians. They used proportions in various practical applications, such as measuring land and constructing buildings. The formalization of cross-multiplication as a technique evolved over centuries as mathematical notation and methods developed.

🔑 Key Principles of Cross-Multiplication

• ⚖️ Proportionality: Understand that proportions express the equivalence of two ratios. If $\frac{a}{b} = \frac{c}{d}$, then the ratios are proportional.
• ✖️ Cross-Multiplication: In the proportion $\frac{a}{b} = \frac{c}{d}$, cross-multiplication involves multiplying $a$ by $d$ and $b$ by $c$. This gives us the equation $ad = bc$.
• 🧮 Solving for Variables: After cross-multiplying, you'll have a simple equation. Use algebraic techniques to isolate and solve for the unknown variable.

➗ Step-by-Step Guide to Solving Proportions with Cross-Multiplication

1. Step 1: Write the Proportion

   Set up the two ratios as a proportion. Ensure that corresponding quantities are in the same positions (numerator or denominator).

2. Step 2: Cross-Multiply

   Multiply the numerator of the first ratio by the denominator of the second ratio, and vice versa. This eliminates the fractions.
3. Step 3: Simplify the Equation

   Simplify both sides of the equation if possible. Combine like terms to make the equation easier to solve.

4. Step 4: Solve for the Variable

   Use algebraic techniques (addition, subtraction, multiplication, division) to isolate the variable you're solving for.

5. Step 5: Check Your Answer

   Substitute the value you found for the variable back into the original proportion to ensure it makes the equation true.

➕ Real-World Examples

Example 1: Recipe Scaling

A recipe calls for 2 cups of flour for every 3 eggs. If you want to use 9 eggs, how much flour do you need?

• 📝 Step 1: Set up the proportion: $\frac{2 \text{ cups flour}}{3 \text{ eggs}} = \frac{x \text{ cups flour}}{9 \text{ eggs}}$
• ✖️ Step 2: Cross-multiply: $2 * 9 = 3 * x$, which simplifies to $18 = 3x$
• ➗ Step 3: Divide both sides by 3: $x = 6$

You need 6 cups of flour.

Example 2: Map Distances

On a map, 1 inch represents 50 miles. If two cities are 3.5 inches apart on the map, what is the actual distance between them?

• 🗺️ Step 1: Set up the proportion: $\frac{1 \text{ inch}}{50 \text{ miles}} = \frac{3.5 \text{ inches}}{x \text{ miles}}$
• ✖️ Step 2: Cross-multiply: $1 * x = 50 * 3.5$, which simplifies to $x = 175$

The actual distance is 175 miles.

📊 Practice Quiz

1. Solve for x: $\frac{4}{7} = \frac{x}{21}$
2. Solve for y: $\frac{5}{8} = \frac{15}{y}$
3. Solve for a: $\frac{a}{6} = \frac{9}{2}$
4. Solve for b: $\frac{3}{b} = \frac{12}{20}$
5. A store sells 3 apples for $2. How much would 9 apples cost?
6. If 2 inches on a map represents 100 miles, how many miles do 5 inches represent?
7. A recipe requires 1 cup of sugar for every 2 cups of flour. How much sugar is needed for 5 cups of flour?

💡 Conclusion

Cross-multiplication is a powerful tool for solving proportions. By understanding the underlying principles and following the step-by-step guide, you can confidently tackle a wide range of proportion problems. Remember to practice regularly to reinforce your skills and build your problem-solving abilities.