Proportions: Finding the Missing Value
A proportion is a statement that two ratios are equal: a/b = c/d. Given any three of the four values, cross-multiplication finds the fourth. Proportional reasoning is one of the most practically useful mathematical skills for everyday life.
Cross-Multiplication Method
a/b = c/d โ a ร d = b ร c
Example: 3/4 = x/28
3 ร 28 = 4 ร x
84 = 4x โ x = 21Real-World Examples
- Recipe scaling: Recipe uses 200g flour for 4 servings. For 10 servings: 200/4 = x/10 โ x = 500g
- Map distance: 1 cm = 25 km. How far is 3.5 cm? โ 87.5 km
- Unit rate: 360 miles in 6 hours. Rate per hour: 360/6 = 60 mph
- Exchange rate: ยฃ1 = $1.27. ยฃ85 = ? โ 85 ร 1.27 = $107.95
Percentage as Proportion
x% of y = ? โ x/100 = ?/y โ ? = xy/100
What % is 15 of 60? โ 15/60 = x/100 โ x = 25%Solve any proportion: Free Proportion Calculator
Direct vs Inverse Proportion
- Direct proportion (y = kx): Double x โ double y. Examples: cost and quantity (buy twice as many, pay twice as much), distance and time at constant speed, Hooke's law (force and spring extension).
- Inverse proportion (xy = k, or y = k/x): Double x โ halve y. Examples: speed and journey time at fixed distance, pressure and volume at constant temperature (Boyle's Law), number of workers and time to complete a task.
Solving Proportion Problems
Cross-multiplication is the core technique: if a/b = c/d, then ad = bc. This allows any one unknown to be solved. Example: a recipe uses 150 g sugar for 12 biscuits. How much for 20 biscuits? Set up: 150/12 = x/20. Cross-multiply: 12x = 3000, x = 250 g. The same method applies to map scale problems: if 1 cm = 5 km and a road is 3.5 cm long on the map, real length = 3.5 ร 5 = 17.5 km.
Frequently Asked Questions
How do I recognise if two quantities are proportional?
Plot them on a graph. Direct proportion gives a straight line through the origin. Inverse proportion gives a hyperbola. Numerically, check the ratio y/x for direct proportion โ if it is constant, they are directly proportional. For inverse, check x ร y โ if constant, they are inversely proportional.
What is the constant of proportionality?
In y = kx, k is the constant of proportionality. It has units: if y is in kg and x is in metres, k has units kg/m. Finding k from one pair of values allows prediction of any other pair: if 3 kg costs ยฃ12, k = ยฃ12/3 = ยฃ4/kg, so 7 kg costs 7 ร ยฃ4 = ยฃ28.
Can you have proportion with more than two quantities?
Yes. Combined variation: y is directly proportional to x and inversely proportional to z, written y = kx/z. Example: electrical resistance R = ฯL/A (resistivity ร length รท cross-section area) โ directly proportional to length and inversely to area. These appear throughout physics and engineering.