Proportions Word Problems

Introduction

Proportions are equations that express the equality of two ratios. In real-world scenarios, we often encounter word problems that can be solved using proportions.

Key Concepts

1. Ratio: A comparison of two quantities.
2. Proportion: An equation stating that two ratios are equal, expressed as $\frac{a}{b} = \frac{c}{d}$.
3. Cross-Multiplication: A method used to solve proportions by multiplying the means and extremes.

Steps to Solve Proportion Word Problems

1. Identify the ratios in the problem.
2. Set up the proportion based on the information given.
3. Cross-multiply to find the unknown value.
4. Solve for the variable.

Example Problem

A recipe requires 2 cups of flour for every 3 cups of sugar. If you have 8 cups of sugar, how much flour do you need?

Step 1: Identify the Ratios

The ratio of flour to sugar is $\frac{2}{3}$.

Step 2: Set Up the Proportion

Let $x$ be the amount of flour needed. Set up the proportion:

$$\frac{2}{3} = \frac{x}{8}$$

Step 3: Cross-Multiply

Cross-multiply to solve for $x$:

$$2 \cdot 8 = 3 \cdot x$$

This simplifies to:

$$16 = 3x$$

Step 4: Solve for $x$

Divide both sides by 3:

$$x = \frac{16}{3} \approx 5.33$$

Thus, you need approximately 5.33 cups of flour.

Key Questions

• What is a ratio and how is it different from a proportion?
• How can you set up a proportion from a word problem?
• What does cross-multiplication help you achieve?

Conclusion

Proportion word problems require careful reading and understanding of relationships between quantities. Practice identifying these relationships to improve your problem-solving skills.