Grade: Grade 6 Subject: Mathematics Unit: Ratios & Rates Lesson: 5 of 6 SAT: Algebra ACT: Math

Application Practice

Learn

In this lesson, you will apply ratio and rate concepts to solve complex real-world problems. These skills are essential for the SAT and ACT, which frequently test your ability to use proportional reasoning.

Real-World Applications of Ratios and Rates

  • Shopping: Comparing prices per unit (unit rates)
  • Cooking: Scaling recipes up or down (equivalent ratios)
  • Travel: Calculating speed, distance, and time
  • Finance: Understanding interest rates and percentages
  • Science: Measuring density, concentration, and scale

Problem-Solving Strategy: RATIO

  • Read the problem carefully - identify what you know and what you need to find
  • Arrange the information - set up a ratio or rate
  • Think about relationships - are the quantities proportional?
  • Implement your plan - use equivalent ratios or unit rates to solve
  • Overify your answer - does it make sense in context?

Key Formulas

Unit Rate: Total Amount / Number of Units

Equivalent Ratios: a/b = c/d (cross multiply: a x d = b x c)

Speed: Distance / Time

Scaling: New Amount = Original Amount x Scale Factor

Examples

Example 1: Best Buy Comparison

Problem: A 24-pack of water bottles costs $4.80. A 36-pack costs $6.48. Which is the better deal?

Solution:

  • 24-pack: $4.80 / 24 = $0.20 per bottle
  • 36-pack: $6.48 / 36 = $0.18 per bottle

The 36-pack is the better deal at $0.18 per bottle.

Example 2: Recipe Scaling

Problem: A recipe for 4 servings calls for 2.5 cups of flour. How much flour is needed for 10 servings?

Solution:

  • Set up the proportion: 2.5 cups / 4 servings = x cups / 10 servings
  • Cross multiply: 2.5 x 10 = 4 x x
  • 25 = 4x
  • x = 6.25 cups

Practice

Solve these application problems using ratios and rates.

Practice 1

A car travels 180 miles in 3 hours. At this rate, how far will it travel in 7 hours?

Practice 2

Store A sells 5 notebooks for $8.75. Store B sells 8 notebooks for $13.60. Which store has the better price per notebook?

Practice 3

A map has a scale of 1 inch = 25 miles. If two cities are 3.5 inches apart on the map, what is the actual distance?

Practice 4

A printer can print 120 pages in 4 minutes. How long will it take to print 450 pages?

Practice 5

Orange juice concentrate requires mixing 3 parts concentrate with 7 parts water. How much water is needed for 15 ounces of concentrate?

Practice 6

In a science experiment, 40 grams of a substance dissolves in 250 mL of water. How many grams will dissolve in 400 mL?

Practice 7

A cyclist rides 15 miles in 45 minutes. At this rate, how long will a 24-mile ride take?

Practice 8

The ratio of boys to girls in a class is 3:5. If there are 24 students total, how many are girls?

Practice 9

A 12-ounce can of soda costs $1.50. A 2-liter bottle (about 67 ounces) costs $2.49. Which is the better value per ounce?

Practice 10

A school survey found that 3 out of every 8 students prefer pizza for lunch. If there are 640 students, how many prefer pizza?

Answers

Click to reveal answers

1. Rate: 180/3 = 60 mph. In 7 hours: 60 x 7 = 420 miles

2. Store A: $8.75/5 = $1.75 each. Store B: $13.60/8 = $1.70 each. Store B is better.

3. 3.5 x 25 = 87.5 miles

4. Rate: 120/4 = 30 pages/min. Time: 450/30 = 15 minutes

5. 3:7 ratio means 3 parts concentrate needs 7 parts water. 15/3 = 5, so 7 x 5 = 35 ounces water

6. 40/250 = x/400. Cross multiply: 40 x 400 = 250x. 16000 = 250x. x = 64 grams

7. Rate: 15 miles/45 min = 1 mile/3 min. Time: 24 x 3 = 72 minutes (1 hour 12 min)

8. 3+5 = 8 parts total. Girls = 5/8 x 24 = 15 girls

9. Can: $1.50/12 = $0.125/oz. Bottle: $2.49/67 = $0.037/oz. The 2-liter bottle is better value.

10. 3/8 x 640 = 240 students prefer pizza

Check Your Understanding

  1. How do you find a unit rate?
  2. What does it mean for two ratios to be equivalent?
  3. How do you use cross multiplication to solve a proportion?
  4. When comparing prices, what should you calculate to find the best deal?
  5. How do you scale a recipe to make more or fewer servings?
Click to check your answers
  1. Divide the total by the number of units to get the amount per one unit.
  2. They represent the same relationship when simplified, and their cross products are equal.
  3. Set up the proportion a/b = c/d, then cross multiply (a x d = b x c) and solve for the unknown.
  4. Calculate the unit price (price per item or per ounce) for each option.
  5. Find the scale factor (new servings / original servings) and multiply each ingredient by that factor.

Next Steps

  • Practice finding unit rates when shopping
  • Try scaling your favorite recipe
  • Complete the Unit Checkpoint to test your mastery