Ratios, rates, percentages & units

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Percentages without gimmicks

• "What is 35% of 80?" → 0.35 × 80 = 28.
• "24 is what percent of 60?" → 24/60 = 0.40 → 40%.
• Increase 250 by 12% → 250 × 1.12 = 280. Decrease by 12% → × 0.88. One multiplication, not two steps.
• Percent change: (new − old)/old × 100. From 40 to 50 is +25%; from 50 to 40 is −20%. Direction matters because the base changes, and this asymmetry is a tested trap.

Ratios and proportions

A ratio of dogs to cats of 3:5 means 3k dogs and 5k cats for some k. Given a real count, find k, then answer anything. For proportions, cross-multiply: x/12 = 7/4 → 4x = 84 → x = 21. Word cue: "at this rate…" almost always means set up a proportion.

Unit conversion: multiply by clever forms of 1

60 miles/hour to feet/second: 60 mi/hr × (5280 ft/mi) × (1 hr/3600 s) = 88 ft/s. Write the fractions so unwanted units cancel diagonally. If the units don't cancel to what the question asks for, the setup is wrong, so fix it before computing.

Common traps

• Percent of the wrong base. A 20% discount then 10% off the discounted price is ×0.8×0.9 = ×0.72, a 28% total discount, not 30%.
• A ratio is not a count. 3:5 doesn't mean 3 dogs exist; it means the multiplier is unknown until a real number appears.
• Per-what confusion. Dollars per pound and pounds per dollar are reciprocals; the units tell you which you have.

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