Ratios, Proportions, and Percent Problems
Key terms, setups, and traps for proportional reasoning with ratios, rates, and percents.
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Questions Covered in This Set
12 cards to master
What is a ratio?
A comparison of two quantities by division, written 12:18, 12/18, or simplified to 2/3. Order matters.
Part-to-part vs. part-to-whole ratio
Part-to-part compares two groups (girls:boys = 2:3); part-to-whole compares one group to the total (girls:students = 12:30 = 2:5).
Ratio of red to blue is 3:4 and there are 28 marbles. How many red?
3 + 4 = 7 parts; 28 ÷ 7 = 4 per part; red = 3 × 4 = 12 (blue = 16).
What is a unit rate, and why use it?
A rate with the second quantity scaled to 1 (120 mi ÷ 4 hr = 30 mph). It makes comparisons like best-buy shopping easy.
How do you solve a proportion a/b = c/d?
Cross multiply: ad = bc, then solve. It's legal because you're multiplying both sides by bd.
Golden rule for setting up a proportion
Keep the same kind of quantity in the same position on both sides (notebooks/dollars = notebooks/dollars), then sanity-check the answer's size.
The one proportion that solves all three percent questions
part/whole = percent/100.
Translate "27 is 45% of what number?" into algebra
27 = 0.45x, so x = 60. ("is" → =, "of" → ×, "what" → x, "percent" → /100)
Multiplier trick for percent increase/decrease
Increase of 20% → multiply by 1.20; decrease of 20% → multiply by 0.80. Chain them: $60 × 0.70 × 1.08 = $45.36.
Percent change formula (and the #1 error)
(new − old)/old × 100%. The error is dividing by the new value instead of the original.
Why doesn't a 20% increase then a 20% decrease return to the start?
The percents apply to different bases: 100 → 120 → 96, a 4% net loss.
After a 25% discount a shirt costs $36. Original price?
$36 is 75% of the original: 0.75x = 36, so x = $48. Don't take 25% of 36.
