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Math 8 & 9 Proportional Reasoning

1. Ratios: Comparing Quantities

A ratio compares two or more quantities with the same units.

  • Part-to-Part: Comparing one group to another (e.g., 5 cats to 3 dogs → 5:3).

  • Part-to-Whole: Comparing one group to the total (e.g., 5 cats to 8 total animals → 5:8). This can be written as a fraction (5/8) or a percent (62.5%).

How to Write a Ratio

  1. Colon: 2:5

  2. Words: 2 to 5

  3. Fraction: 2/5

Rule: Always simplify ratios. Divide both sides by the Greatest Common Factor (GCF). Example: 10:15 simplifies to 2:3 (both divided by 5).

2. Rates: Comparing Different Units

A rate compares two quantities with different units. You must always include the units.

  • Example: Driving 200 km in 4 hours (200 km / 4 hrs).

  • Example: Earning $60 for 5 hours of work ($60 / 5 hrs).

3. Unit Rates & Unit Prices

A unit rate is a rate where the second quantity is 1. To find it, divide the first number by the second.

The "Best Buy" Formula (Unit Price)

To find out which item is the better deal, calculate the cost for one unit.

  • Formula: Unit Price = Price ÷ Quantity

Comparison Example:

  • Option A: $4.50 for 3 apples → ($4.50 ÷ 3) = $1.50 per apple

  • Option B: $10.00 for 8 apples → ($10.00 ÷ 8) = $1.25 per apple

  • Winner: Option B is the "Best Buy" because the unit price is lower.

4. Solving Proportions (Cross-Multiplication)

A proportion is an equation showing that two ratios or rates are equal. Use the "Cross-Multiply and Divide" method to find a missing value.

Example Problem: If 3 kg of oranges cost $7.50, how much do 5 kg cost?

  1. Set up the equation: $7.50 / 3 kg = x / 5 kg

  2. Cross-multiply: 3 * x = 7.50 * 5

  3. Calculate: 3x = 37.50

  4. Solve for x: x = 37.50 ÷ 3 = $12.50

5. Quick Review Checklist

  • [ ] Is the ratio simplified to its lowest terms?

  • [ ] Did I include units for my rates (e.g., $/kg, km/h)?

  • [ ] For "Best Buy" questions, is the Price the first number in the division? (Price ÷ Amount)

  • [ ] In a proportion, are the units in the same spots on both sides? (e.g., Liters over Dollars = Liters over Dollars)

 

PRACTICE PROBLEMS: Proportional Reasoning

Section 1: Ratios & Rates Warm-up

  1. Simplify: A recipe calls for 12 cups of flour and 8 cups of sugar. What is the simplest ratio of flour to sugar?

  2. Part-to-Whole: In a class of 30 students, 12 are boys. What is the ratio of boys to the total number of students in simplest form?

  3. Basic Rate: A cyclist travels 45 km in 3 hours. What is their unit rate in km/h?

  4. Simplifying Ratios: Write the ratio 150:250 in its simplest form.

Section 2: The "Best Buy" Challenge

Find the unit price for each and determine which is the better deal.

  1. Laundry Detergent:

    • Brand A: 1.5 L for $6.45

    • Brand B: 4 L for $16.80

  2. Juice Boxes:

    • Pack of 10 for $4.50

    • Pack of 24 for $10.32

  3. Snack Mix:

    • 250g bag for $3.75

    • 600g bag for $8.40

Section 3: Solving Proportions

Use cross-multiplication to find the missing value (x).

  1. Map Scale: On a map, 2 cm represents 50 km. If two cities are 7.5 cm apart on the map, how many kilometers apart are they in real life?

  2. Recipe Scaling: A recipe for 4 people requires 3 eggs. How many eggs do you need if you are cooking for 14 people?

  3. The Mystery Number: Solve for $x$ in the following proportion:

    4.5 / 12 = x / 40

ANSWER KEY (For Self-Checking)

  1. 3:2 (Divide both by 4)

  2. 2:5 (12:30 simplifies by dividing both by 6)

  3. 15 km/h (45 ÷ 3)

  4. 3:5 (Divide both by 50)

  5. Brand B (Brand A: $4.30/L vs. Brand B: $4.20/L)

  6. Pack of 24 (10-pack: $0.45/box vs. 24-pack: $0.43/box)

  7. 250g bag (Small: $0.015/g vs. Large: $0.014/g — Wait, check that! Small: 1.5 cents/g vs Large: 1.4 cents/g. Large is actually better.)

  8. 187.5 km (2x = 375)

  9. 10.5 eggs (4x = 42. In a real kitchen, you'd use 11!)

  10. x = 15 (12x = 180)

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