Ratios, Proportions, and Percent
Grade 9 · Math · Study Guide
Solve proportions using cross-multiplication, apply percent change, and use ratios to model real-world situations.
A ratio compares two quantities using division. The ratio 3:5 can also be written as 3/5 or '3 to 5.' Ratios are used to scale recipes, maps, and models. A proportion states that two ratios are equal: a/b = c/d. To solve for a missing value, use cross-multiplication: a × d = b × c.
Percent means 'per hundred.' To find a percent of a number, convert the percent to a decimal and multiply: 30% of 80 = 0.30 × 80 = 24. Percent change measures how much a quantity increases or decreases relative to its original value: percent change = (new − old) / old × 100%. If a price increases from $50 to $65, the percent increase = 15/50 × 100% = 30%.
Unit rates express a ratio with a denominator of 1 and are extremely useful in everyday life — miles per hour, price per ounce, words per minute. Scale factors in similar figures use proportions to find unknown lengths.
| Percent | Decimal | Fraction |
|---|---|---|
| 25% | 0.25 | 1/4 |
| 50% | 0.50 | 1/2 |
| 75% | 0.75 | 3/4 |
| 10% | 0.10 | 1/10 |
| 33.3% | 0.333... | 1/3 |
Key terms
- Ratio
- A comparison of two quantities by division; written a:b or a/b
- Proportion
- An equation stating two ratios are equal: a/b = c/d
- Cross-multiplication
- In a proportion a/b = c/d, multiply diagonally: ad = bc
- Percent
- Parts per hundred; convert to decimal by dividing by 100
- Percent change
- (New − Original) / Original × 100%
- Unit rate
- A ratio with a denominator of 1; e.g., 60 miles per 1 hour
Flashcards
- Ratio
- Comparison of two quantities: a:b or a/b
- Proportion
- Two equal ratios: a/b = c/d
- Cross-multiplication
- ad = bc; used to solve proportions
- Percent
- Out of 100; move decimal two places left to convert to decimal
- Percent change
- (New − Original) / Original × 100%
- Unit rate
- A rate with a denominator of 1; e.g., $3.50 per pound
Practice questions
Solve the proportion: x/5 = 12/20
- x = 2
- x = 3
- x = 4
- x = 6
Show answer
Answer: x = 3
Cross-multiply: 20x = 60 → x = 3.
A 20% decrease on a $100 item brings the price to $80.
Show answer
Answer: True
20% of $100 = $20; $100 − $20 = $80.
40% of 200 is ___.
Show answer
Answer: 80
0.40 × 200 = 80.
A price rose from $40 to $50. What is the percent increase?
- 10%
- 20%
- 25%
- 40%
Show answer
Answer: 25%
Percent increase = (50 − 40)/40 × 100% = 10/40 × 100% = 25%.
A car travels 300 miles on 10 gallons of gas. What is the unit rate (miles per gallon)?
- 20
- 25
- 30
- 35
Show answer
Answer: 30
300 miles ÷ 10 gallons = 30 miles per gallon.
Fun facts
- The word 'percent' comes from the Latin 'per centum' meaning 'by the hundred' — the same root as century (100 years) and cent (1/100 of a dollar).
- When banks advertise 'APR' (annual percentage rate), they are using percent to describe how much interest you owe or earn per year.
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