Ratio & Proportion Guide
A ratio compares two quantities. A proportion states that two ratios are equal. Together, these concepts underpin much of practical mathematics: cooking, construction, maps, finance, and science all rely on proportional reasoning.
This guide covers what ratios and proportions are, how to solve any proportion using cross-multiplication, and how to apply them to real-world problems: unit rates, map scales, recipe scaling, and speed/distance/time.
Formula
a/b = c/d ⟹ a × d = b × c (cross-multiplication)
What is a Ratio?
A ratio compares two (or more) quantities. The ratio 3:4 means "for every 3 of the first, there are 4 of the second." Ratios can be written as 3:4, 3/4, or "3 to 4."
Part-to-part ratio: In a class of 12 boys and 18 girls, the boy-to-girl ratio is 12:18 = 2:3 (simplified). This compares the two groups to each other.
Part-to-whole ratio: In the same class, boys represent 12/30 = 2/5 = 40% of the class. This compares one group to the total.
Simplifying ratios: Divide all parts by their GCD. 12:18 → GCD(12,18) = 6 → 2:3. For 15:25:40 → GCD = 5 → 3:5:8.
Three-part ratios: Concrete mix cement:sand:gravel = 1:2:4. For 700 kg total: cement = 700×(1/7) = 100 kg, sand = 200 kg, gravel = 400 kg.
What is a Proportion?
A proportion is an equation stating that two ratios are equal: a/b = c/d. Proportions model situations where quantities scale together — if one doubles, the other doubles.
Direct proportion: y = kx (y is proportional to x). If 3 apples cost $2.40, then 7 apples cost x: 3/2.40 = 7/x → x = (7 × 2.40) / 3 = $5.60.
Inverse proportion: y × x = k (y inversely proportional to x). If 4 workers finish in 6 days, 8 workers finish in: 4×6 = 8×d → d = 3 days.
Checking a proportion: 3/4 = 9/12? Cross-multiply: 3×12 = 36 and 4×9 = 36. Equal, so yes — it is a valid proportion.
Cross-Multiplication — Solving for the Unknown
Given a/b = c/d, cross-multiply to get: a × d = b × c. To find any missing term:
Find d: d = (b × c) / a. Find c: c = (a × d) / b. Find a: a = (b × c) / d. Find b: b = (a × d) / c.
Example: If 5/8 = x/40, then 5 × 40 = 8 × x → 200 = 8x → x = 25.
Word problem: A recipe uses 3 cups of flour for 24 cookies. How much flour for 60 cookies? 3/24 = x/60 → x = (3 × 60) / 24 = 180/24 = 7.5 cups.
| Proportion | Cross Products | Solution |
|---|---|---|
| 1/4 = x/20 | 1×20 = 4×x | x = 5 |
| x/6 = 15/9 | x×9 = 6×15 | x = 10 |
| 7/x = 14/6 | 7×6 = x×14 | x = 3 |
| 3/5 = 12/x | 3×x = 5×12 | x = 20 |
| x/4 = 9/12 | x×12 = 4×9 | x = 3 |
Unit Rate
A unit rate expresses a ratio with denominator 1. "60 km in 1.5 hours" → unit rate = 60/1.5 = 40 km/h.
Comparison shopping: Brand A: 400g for $3.20 → $3.20/400 = $0.008/g. Brand B: 650g for $4.55 → $4.55/650 = $0.007/g. Brand B is cheaper per gram.
Wage rate: Earned $247 in 13 hours → $247/13 = $19/hour.
Mileage: Car travels 285 miles on 9.5 gallons → 285/9.5 = 30 miles per gallon (mpg).
Unit rates allow direct comparison between quantities with different totals. Always divide the numerator by the denominator to find "per 1 unit" of the denominator.
Scale and Maps
A map scale is a ratio: map distance : real distance. Scale 1:50,000 means 1 cm on map = 50,000 cm = 500 m = 0.5 km in reality.
Example: Two cities are 7.4 cm apart on a 1:200,000 scale map. Real distance = 7.4 × 200,000 cm = 1,480,000 cm = 14.8 km.
Finding map distance from real distance: Cities are 35 km apart. On a 1:500,000 scale map: 35 km = 3,500,000 cm. Map distance = 3,500,000 / 500,000 = 7 cm.
Scale models: A model car at 1:18 scale is 24 cm long. Real car length = 24 × 18 = 432 cm = 4.32 m.
Architectural drawings: Scale 1:100. A room shown as 4.5 cm × 3.2 cm is actually 450 cm × 320 cm = 4.5 m × 3.2 m.
Recipe Scaling
Recipe scaling uses direct proportion. If a recipe for 4 serves needs 200g flour, scaling to 10 serves: 200/4 = x/10 → x = 500g.
Scale factor method: Scale factor = (desired servings) / (original servings). For 4 → 10: factor = 10/4 = 2.5. Multiply every ingredient by 2.5.
Example: Cake recipe for 8: flour 300g, sugar 200g, butter 150g, eggs 4. Scale to 20 serves (factor = 20/8 = 2.5): flour 750g, sugar 500g, butter 375g, eggs 10.
Note on scaling: Cooking times and temperatures do NOT scale linearly — a larger cake may need more time but not proportionally more. Leavening agents (baking powder) often need less than proportional increase to avoid over-rising.
Speed, Distance, and Time as Proportions
The fundamental relationship: Distance = Speed × Time (D = S × T). Rearranging: S = D/T and T = D/S.
Example 1: A train travels at 120 km/h for 2.5 hours. Distance = 120 × 2.5 = 300 km.
Example 2: Distance 240 miles, time 4 hours. Speed = 240/4 = 60 mph.
Example 3: Distance 150 km at 60 km/h. Time = 150/60 = 2.5 hours = 2 hours 30 minutes.
Proportion approach: If a car covers 90 km in 1 hour, how long for 225 km? 90/1 = 225/t → t = 225/90 = 2.5 hours.
Average speed: Total distance / Total time. 300 km in first 3 hours + 200 km in next 2 hours. Average = 500/5 = 100 km/h. (Do NOT average the two individual speeds directly.)