Ratios, Proportions, and Rates

ACT Math · Axiom Academy

LESSON Ratios, Proportions, and Rates One relationship, four disguises — comparing quantities, solving proportions, and reading rates as motion 1. A Ratio Is a Comparison, Not a Count A ratio compares two (or more) quantities by division. Fruit in a 2:3:4 ratio of apples : oranges : bananas doesn't mean exactly 2, 3, and 4 pieces — it means for every 2 apples there are 3 oranges and 4 bananas, no matter how large the stand gets. Watch all three bars grow together, keeping the same ratio at every step. Scaling every part by the same factor k preserves the ratio 2. Solving a Proportion: Cross-Multiplication in Motion A proportion is an equation stating that two ratios are equal: , read " a is to b as c is to d ." Watch the two diagonal products build across the equals sign — that's the entire cross-multiplication rule, made visible. The cross-product rule: multiply diagonally across the equals sign 1 inch represents 50 miles. Two cities are 3.5 inches apart on the map: , so x = 50(3.5)=175 miles. 4 people need 2 cups of flour. For 10 people: , so 4x = 20 and x = 5 cups. 3. Unit Rates: The Line's Steepness IS the Speed A unit rate compares a quantity to exactly 1 unit of another — miles per 1 hour, dollars per 1 pound, words per 1 minute. On a distance-vs-time graph, the unit rate is the line's slope: . Watch two trains leave the same station in opposite directions, at 55 and 65 mph — the steeper line is the faster train, and the gap between them grows at their COMBINED rate.

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