A ratio compares two quantities, like 3:5 or 3⁄5. Ratios can compare things with the same units (like 3 boys to 5 girls) or different units (like 60 km per hour). A rate is a special ratio that compares different units. For example, "60 kilometres per hour" is a rate. A unit rate simplifies the comparison to "per 1 unit," making it easy to compare options. For instance, $2.50 per 100 g becomes $0.025 per gram.
Proportional relationships keep the same ratio throughout. If you double one quantity, the other doubles too. Graphically, proportional relationships form straight lines through the origin and follow y = k·x, where k is the constant of proportionality (the unit rate). For example, if apples cost $3 per kilogram, then 2 kg costs $6, 3 kg costs $9, and so on — the ratio of cost to weight stays constant at 3.
Percents are a special kind of ratio meaning "per 100." They let you compare parts to wholes in a standard way. Converting between percent, decimal, and fraction is essential: percent → decimal by dividing by 100 (18% = 0.18), decimal → percent by multiplying by 100 (0.045 = 4.5%), and percent of a number by multiplying (30% of 80 → 0.30 × 80 = 24).
Percent change measures how much something grows or shrinks relative to its original value. The formula is: Percent Change = (New − Original)⁄Original × 100%. A markup increases the original price, while a discount decreases it. Taxes and tips are also percent calculations. Proportions are equations stating two ratios are equal: a⁄b = c⁄d. You can solve them using cross-multiplication (a·d = b·c) or by scaling both sides.
Real-world connection: Unit rates help you find the best deal at the grocery store. Percent change helps you understand salary raises and sale discounts. Proportions help architects scale blueprints and cooks adjust recipes.
Unit rate comparison — which is the better deal?
| Option | Total Cost | Quantity | Unit Rate |
|---|---|---|---|
| Option A | $4.80 | 12 cookies | $0.40 per cookie |
| Option B | $7.20 | 20 cookies | $0.36 per cookie |
Percent change examples
| Situation | Original | New | Change | % Change |
|---|---|---|---|---|
| Price increase | $40 | $50 | +$10 | +25% |
| Price decrease | $80 | $68 | −$12 | −15% |
| Population growth | 24,000 | 25,200 | +1,200 | +5% |
Proportional relationship table — cost of apples at $3/kg
| Weight (kg) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Cost ($) | 0 | 3 | 6 | 9 | 12 | 15 |