Direct and Inverse Proportion
This topic covers how variables change in relation to one another. Understanding direct and inverse proportion is crucial for setting up and solving equations that model real-world relationships, a common task in science and engineering problems.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- Direct proportion (y ∝ x) means y = kx. If x doubles, y doubles. The graph is a straight line passing through the origin.
- Inverse proportion (y ∝ 1/x) means y = k/x. If x doubles, y is halved. The graph is a rectangular hyperbola.
- Proportion can involve powers, such as y ∝ x2 or y ∝ √(x) (y = k × x^(1/2)). The change in y depends on the change in x raised to that power.y = kx2
- Solving any proportion problem is a two-step process: first, use a given pair of values to calculate the constant of proportionality, k. Second, use this value of k to find the unknown quantity.
- A relationship y = ax + b where b is not zero is a linear relationship, but it is NOT direct proportion because the graph does not pass through the origin and y/x is not constant.
Diagram
Formulae
y = k × x When y is directly proportional to x.
y = k / x When y is inversely proportional to x.
y = k × xn When y is proportional to x raised to the power of n. This general form covers direct (n=1), inverse (n=-1), square (n=2), square root (n=1/2), etc.
Definitions
- Direct Proportion
- A relationship between two variables where their ratio is a constant. As one increases, the other increases by the same factor.
- Inverse Proportion
- A relationship between two variables where their product is a constant. As one increases, the other decreases by the same factor.
- Constant of Proportionality (k)
- The fixed, non-zero number that links the two variables in a proportional relationship. It is the 'k' in equations like y = kx and y = k/x.
Worked example
The electrical resistance, R, of a wire is inversely proportional to the square of its diameter, d. A wire with a diameter of 2 mm has a resistance of 3 Ohms. What is the resistance of a wire of the same material and length with a diameter of 3 mm?
- 1
Step 1:
Write down the algebraic relationship.
'R is inversely proportional to the square of d' means R ∝ 1/(d2), so R = k / (d2).
- 2
Step 2:
Use the given values to find the constant, k.
Substitute R = 3 and d = 2 into the equation:
3 = k / (22), so 3 = k / 4 - 3
Step 3:
Solve for k.
Multiply both sides by 4 to get k = 12.
- 4
Step 4:
Write the full equation:
R = 12 / (d2) - 5
Step 5:
Use the full equation to find the new resistance.
Substitute d = 3:
R = 12 / (32) = 12 / 9 - 6
Step 6:
Simplify the fraction.
R = 12 / 9 = 4 / 3 Ohms
Answer: 4/3 Ohms
Common mistakes
- ×Incorrect power usage: A common mistake is using 'x' when the relationship is with 'x2' or '√(x)'. For instance, if 'y is proportional to the square of x', using y=kx instead of y=kx2 will lead to an answer that is off by a factor.
- ×Confusing direct and inverse: Setting up the equation as y = kx when it should be y = k/x. Always double-check if one variable should increase or decrease as the other increases.
- ×Ratio-scaling errors: When scaling from one scenario to another, failing to apply the power to the scaling factor. If d doubles in R ∝ 1/d2, R changes by a factor of 1/(22) = 1/4, not 1/2.
- ×Arithmetic errors with fractional powers: Miscalculating expressions like 8^(2/3). This should be calculated as (cuberoot(8))2 = 22 = 4, not as a decimal multiplication.
No-calculator tips
- ✓Keep k as a fraction. If you find that 10 = k × 4, it's often better to work with k = 10/4 = 5/2 than to use the decimal 2.5, as fraction multiplication can be simpler.
- ✓For fractional powers like x^(a/b), always do the 'b' root first, then the 'a' power. For example, to calculate 27^(2/3), find the cube root of 27 first (which is 3), then square it (32 = 9). This keeps the numbers smaller.
- ✓Look for cancellations. In the worked example, R = 12/9. Both numbers are divisible by 3, simplifying the answer to 4/3. Don't leave answers unsimplified.