Lecture Notes: Ratios and Proportions

Introduction

• Topic: Solving problems involving ratios and proportions.

Example 1: Ratio of Cats to Dogs

• Problem: Island with 540 cats and 675 dogs.
• Objective: Find the ratio of cats to dogs.
• Steps:
  • Set up a fraction: cats/dogs = 540/675.
  • Simplify the fraction:
    • Divide by 5: 540/5 = 108, 675/5 = 135.
    • Divide by 9: 108/9 = 12, 135/9 = 15.
    • Further simplify: 12 = 4x3, 15 = 5x3. Cancel 3, resulting in 4/5.
• Result: The ratio of cats to dogs is 4 to 5 (4:5).

Example 2: Ratio of Boys to Girls

• Problem: Ratio of boys to girls is 8:7, with 40 boys.
• Objective: Find the number of girls.
• Steps:
  • Set up proportion: 8/7 = 40/x.
  • Cross-multiply: 8x = 40x7.
  • Solve: 8x = 280, x = 35.
• Result: There are 35 girls.
• Quick Calculation:
  • Multiply both sides by factor to expand ratio (8 to 40, multiply by 5).
  • Girls: 7x5 = 35.

Example 3: Cakes Made in Given Hours

• Problem: Karen makes 14 cakes in 6 hours. How many in 15 hours?
• Objective: Find the number of cakes in 15 hours.
• Steps:
  • Set up proportion: 14/6 = x/15.
  • Cross-multiply: 6x = 14x15.
  • Solve: 6x = 210, x = 35.
• Result: She can make 35 cakes in 15 hours.
• Quick Calculation:
  • Ratio 14:6, multiply both sides by 2.5 (15/6 = 2.5).
  • Cakes: 14x2.5 = 35.

Example 4: Rectangle Dimensions

• Problem: Small rectangle: length = 9 inches, width = 8 inches; Large rectangle length = 24 inches.
• Objective: Find width of large rectangle.
• Steps:
  • Set up proportion: 9/8 = 24/x.
  • Cross-multiply: 9x = 8x24.
  • Solve: 9x = 192, x = 64/3 (exact), approx. 21.3 inches.
• Result: Width of large rectangle is 64/3 inches (approx. 21.3 inches).

Example 5: Coin Jar Problem

• Problem: Ratio of nickels, dimes, and quarters is 3:4:7 with total 112 coins.
• Objective: Find number of nickels.
• Steps:
  • Ratios: 3 (nickels) + 4 (dimes) + 7 (quarters) = 14 total.
  • Set fraction: 3/n = 14/112.
  • Cross-multiply: 14n = 3x112.
  • Solve: 14n = 336, n = 24.
• Result: 24 nickels.
• Additional Calculation:
  • Total coins: 14x8 = 112.
  • Dimes: 4x8 = 32, Quarters: 7x8 = 56.
  • Total: 24 (nickels) + 32 (dimes) + 56 (quarters) = 112 coins.

Conclusion

• Various methods and steps are used to solve ratio and proportion problems, each adaptable to different scenarios.