Ratios and Proportions (Introduction)
Learn to compare two quantities using ratios and scale them up or down using proportions — the bridge from arithmetic to algebra.
Ratios power everything from recipe scaling to medication dosing to currency exchange. A proportion is how you take a known relationship and apply it at any scale. These skills bridge everyday arithmetic directly into algebra and real-world problem solving.
- Write and interpret ratios in all three notations: a:b, a/b, and "a to b"
- Set up and solve a basic proportion to find a missing value
- Apply proportional reasoning to scaling, unit rates, and real-world problems
A ratio compares two quantities and tells you their relationship. A proportion says two ratios are equivalent — which lets you find unknown values by scaling one ratio to match another.
\[ \frac{a}{b} = \frac{c}{d} \implies a \times d = b \times c \]Unit rates make comparison easy: if one store charges $1.20 for 4 oz and another charges $1.80 for 6 oz — find the per-oz rate for each to compare fairly.
Ratio Scaling Table
Recipe scaling: 2 cups flour for every 4 servings
| Servings | Cups of Flour | Ratio | Simplified |
|---|---|---|---|
| 4 | 2 | 2:4 | 1:2 |
| 8 | 4 | 4:8 | 1:2 |
| 10 | 5 | 5:10 | 1:2 |
The ratio always simplifies to 1:2 — 1 cup of flour per 2 servings. This is the unit rate.
A bag contains 3 blue marbles and 5 red marbles. Write the ratio of blue to red.
- As a ratio: 3:5
- As a fraction: \(\frac{3}{5}\)
- In words: 3 to 5
Note: 3:5 is not the same as 5:3. Order always matters in a ratio.
A recipe for 4 people uses 2 cups of flour. How much flour for 10 people?
\[ \frac{2 \text{ cups}}{4 \text{ people}} = \frac{x \text{ cups}}{10 \text{ people}} \]Cross-multiply: \(2 \times 10 = 4 \times x\) → \(20 = 4x\) → \(x = 5\)
You need 5 cups of flour for 10 people.
A liquid medication contains 50mg in every 2mL. A patient needs 125mg. How many mL?
\[ \frac{50 \text{ mg}}{2 \text{ mL}} = \frac{125 \text{ mg}}{x \text{ mL}} \]Cross-multiply: \(50x = 125 \times 2 = 250\) → \(x = 5\)
Administer 5mL of the medication.
Try these:
- Write the ratio of 6 cats to 10 dogs in simplest form.
- If 3 tickets cost $24, how much do 7 tickets cost? (Set up a proportion.)
- A car travels 150 miles on 5 gallons. What's the unit rate (miles per gallon)?
▶ Show Answers
- 3:5 — divide both by 2.
- $56 — 3/24 = 7/x → x = (24×7)/3 = 56.
- 30 miles per gallon — 150 ÷ 5 = 30 mpg.
- Reversing the ratio: 3 cats to 5 dogs (3:5) is not the same as 5:3. The order you write a ratio must match the order described in the problem.
- Dropping units: A unit rate without units is meaningless. "30 per" tells you nothing. Always include the units: 30 miles per gallon, $0.35 per bottle.
- Cross-multiplying incorrectly: In \(\frac{a}{b} = \frac{c}{d}\), you multiply \(a \times d\) and \(b \times c\) — not \(a \times c\) or \(b \times d\).
- A ratio compares two quantities — order matters, always.
- A proportion sets two ratios equal — use it to scale or find missing values.
- Unit rates simplify comparison — reduce any ratio to "per 1" for easy side-by-side analysis.
- Cross-multiply to solve: if \(\frac{a}{b} = \frac{c}{d}\) then \(a \times d = b \times c\).
Healthcare workers calculate drug doses using proportions every shift. Engineers use scale ratios to translate blueprint dimensions into real measurements. Currency traders convert between currencies at real-time exchange ratios. Proportional reasoning is one of the most universally applied skills in any technical or analytical career.
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Ratios and Proportions (Introduction) — Quiz
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