Lesson 56: Multi-Step Ratio & Percent Problems

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• Lesson Description
• Learning Objectives
• Topics Covered
• Key Terms
• Lesson Examples
• Practice Problems
• Tips & Common Mistakes
• Summary
• Challenge Problems
• Unit Navigation

Lesson Description

This lesson focuses on solving multi-step problems that combine ratios, proportions, and percent calculations. Students will learn to break problems into manageable steps and apply appropriate formulas in sequence.

Learning Objectives

• Solve problems involving multiple ratios or proportions in sequence.
• Apply percent calculations within multi-step scenarios.
• Integrate ratio, proportion, and percent concepts to real-world problems.

Topics Covered

• Multi-step ratio problems
• Multi-step percent problems
• Combined ratio and percent scenarios
• Stepwise problem-solving strategies

Key Terms

• Ratio: A comparison of two quantities.
• Proportion: An equation stating that two ratios are equal.
• Percent: A ratio expressed as a fraction of 100.
• Multi-Step Problem: A problem that requires more than one calculation to reach a solution.

Lesson Examples

Example 1: Multi-Step Ratio Problem

1. The ratio of boys to girls in a class is 3:4. There are 21 boys. How many girls are there?
2. Set up proportion: 3/4 = 21/x
3. Cross-multiply: 3x = 84 → x = 28 girls

Example 2: Multi-Step Percent Problem

1. A shirt costs $50. It is on sale for 20% off, then a 10% sales tax is added. What is the final price?
2. Discount: 20% of 50 = $10 → discounted price = 50 - 10 = $40
3. Tax: 10% of 40 = $4 → final price = 40 + 4 = $44

Practice Problems

1. A recipe uses a ratio of 2 cups flour to 3 cups sugar. If you use 6 cups flour, how much sugar is needed?
2. A $200 jacket is on sale for 15% off, and a 5% sales tax applies. Calculate the final price.
3. The ratio of adults to children on a trip is 5:2. There are 35 adults. How many children?
4. A store increases prices by 10%, then offers a 5% discount. What is the net percent change?

Tips & Common Mistakes

• Always handle one step at a time; do not combine calculations prematurely.
• Convert percentages to decimals before multiplying.
• Check each step to ensure proportional relationships are maintained.

Summary

Multi-step ratio and percent problems require careful sequencing of calculations. Breaking the problem into steps, converting percentages appropriately, and checking proportions ensures accurate solutions.

Challenge Problems

1. A mixture contains milk and water in a ratio of 5:2. If 14 liters of water are added, what is the new ratio?
2. A product costs $120. It is discounted 25% and then taxed 8%. What is the final price?
3. The ratio of red to blue marbles is 7:3. If 21 red marbles are added, how many blue marbles should be added to keep the same ratio?