Direct and Inverse Proportion
What Is Proportion?
Two quantities are in proportion when they are related in a consistent, predictable way. There are two main types: direct and inverse.
Understanding proportion lets you solve problems like scaling a recipe, calculating speed, working out pay rates, and analysing scientific experiments.
Direct Proportion
Two variables are in direct proportion when they increase or decrease at the same rate. If one doubles, the other doubles. If one is cut in half, the other is also cut in half.
The notation is: y β x (read: "y is proportional to x")
This relationship is written as an equation using a constant of proportionality, k:
y = kx
The value of k tells you exactly how many times bigger y is than x.
Finding k
To find k, substitute a known pair of values into y = kx and solve.
Example: A car travels at constant speed. It covers 120 km in 2 hours. Find the distance after 5 hours.
- d = kt (distance is directly proportional to time)
- 120 = k Γ 2 β k = 60
- After 5 hours: d = 60 Γ 5 = 300 km
Here k = 60 is the speed in km/h. The constant of proportionality often has real meaning.
Graph of Direct Proportion
Direct proportion always produces a straight line through the origin (0, 0). The gradient of the line equals k.
If you plot the pairs and the line does not pass through the origin, the relationship is not direct proportion β it may be a linear relationship, but with a y-intercept, not true proportionality.
Key check: if x = 0, then y = k Γ 0 = 0. The origin is always on the line.
Inverse Proportion
Two variables are in inverse proportion (or inversely proportional) when one increases as the other decreases, such that their product is always the same.
The notation is: y β 1/x
Written as an equation:
y = k/x (equivalently, xy = k)
Finding k for Inverse Proportion
Example: 6 workers take 8 days to complete a project. How many days would 4 workers take (assuming they all work at the same rate)?
- d = k/w (days is inversely proportional to number of workers)
- k = d Γ w = 8 Γ 6 = 48
- With 4 workers: d = 48/4 = 12 days
Notice that as the number of workers goes down, the number of days goes up β which matches what we'd expect.
Graph of Inverse Proportion
Inverse proportion produces a hyperbola β a curve that gets closer and closer to both axes but never touches them. As x increases, y decreases rapidly at first, then more slowly. As x approaches zero, y grows without bound.
The curve is always in the first quadrant when both x and y are positive.
Comparing the Two Types
| Feature | Direct Proportion | Inverse Proportion |
|---|---|---|
| Equation | y = kx | y = k/x |
| Notation | y β x | y β 1/x |
| What's constant | y/x = k | xy = k |
| Graph shape | Straight line through origin | Hyperbola |
| When x doubles... | y doubles | y halves |
Other Proportional Relationships
Proportion isn't limited to simple linear or reciprocal relationships. You may also encounter:
y β xΒ² β y is proportional to the square of x, so y = kxΒ²
Example: The stopping distance of a car is proportional to the square of its speed. If a car travelling at 30 mph stops in 9 m, find the stopping distance at 60 mph.
- d = ksΒ² β 9 = k Γ 30Β² β 9 = 900k β k = 0.01
- At 60 mph: d = 0.01 Γ 60Β² = 0.01 Γ 3600 = 36 m
Doubling the speed quadruples the stopping distance β an important real-world result.
y β 1/xΒ² β y is proportional to the inverse square of x, so y = k/xΒ²
This appears in physics: the intensity of light and the gravitational force both follow inverse square laws.
Solving Proportion Problems: A Step-by-Step Method
- Identify the type of proportion (direct, inverse, or other).
- Write the proportionality statement: y β x or y β 1/x (or y β xΒ², etc.).
- Convert to an equation: introduce the constant k (y = kx or y = k/x, etc.).
- Substitute the known values to find k.
- Use the equation to answer the question.
Worked example: The cost C of printing leaflets is directly proportional to the number n printed. 200 leaflets cost Β£15.
Find: (a) the cost of 500 leaflets, (b) how many leaflets you can print for Β£36.
- C = kn
- 15 = k Γ 200 β k = 0.075
- Equation: C = 0.075n
(a) C = 0.075 Γ 500 = Β£37.50
(b) 36 = 0.075n β n = 36 Γ· 0.075 = 480 leaflets
Common Mistakes
Confusing direct and inverse proportion. "More workers β fewer days" is inverse, not direct. Ask yourself: as one goes up, does the other go up (direct) or down (inverse)?
Forgetting to find k first. Many students try to scale values directly without writing the equation. This works for simple cases but fails with squared or square-rooted relationships. Always find k explicitly.
Using the wrong relationship. If a question says "proportional to the square of x", the equation is y = kxΒ², not y = kx. Read carefully.
Assuming proportionality when there is an intercept. If two quantities are related by y = 3x + 5, they are not in direct proportion β the graph does not pass through the origin.
Real-World Examples
Proportion appears everywhere once you know what to look for:
- Pay and hours worked β straight wages are directly proportional to hours (y = kx where k is the hourly rate).
- Speed and journey time β at a fixed distance, time is inversely proportional to speed.
- Recipe scaling β ingredient quantities are directly proportional to the number of servings.
- Gravity and distance β gravitational force is inversely proportional to the square of the distance between objects.
- Fuel and distance β fuel consumed is directly proportional to distance driven (at constant speed).
Recognising which type of proportion applies in a given context β and being able to find k from a single data point β is the core skill this topic builds.