Direct and Inverse Proportion (Algebraic)

This lesson covers the algebraic treatment of direct and inverse proportion, primarily aimed at Higher tier students in the Edexcel GCSE Mathematics (1MA1) specification. You will learn to set up proportionality equations, find constants, and use them to solve problems. Graphical representations of proportional relationships are also covered.

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Key Vocabulary

Symbol/Term	Meaning
y is proportional to x	$y = kx$y=kx (written $y \propto x$y∝x)
k	The constant of proportionality
∝	The proportionality symbol (means "is proportional to")
Direct proportion	As one variable increases, the other increases at the same rate
Inverse proportion	As one variable increases, the other decreases

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Direct Proportion

If y is directly proportional to x, we write:

$y \propto x$y∝x, which means $y = kx$y=kx

To find k, substitute a known pair of values.

$y \propto x$y∝x (y is directly proportional to x)

Worked Example 1

y is directly proportional to x. When x = 4, y = 20. Find the equation and the value of y when x = 7.

1. $y = kx$y=kx
2. 20 = k x 4, so k = 5
3. Equation: $y = 5x$y=5x
4. When x = 7: y = 5 x 7 = 35

$y \propto x^2$y∝x2 (y is directly proportional to x squared)

Worked Example 2

y is directly proportional to x^2. When x = 3, y = 36. Find y when x = 5.

1. $y = kx^2$y=kx2
2. 36 = k x 9, so k = 4
3. Equation: $y = 4x^2$y=4x2
4. When x = 5: y = 4 x 25 = 100

$y \propto x^3$y∝x3 (y is directly proportional to x cubed)

Worked Example 3

y is directly proportional to x^3. When x = 2, y = 24. Find y when x = 5.

1. $y = kx^3$y=kx3
2. 24 = k x 8, so k = 3
3. Equation: $y = 3x^3$y=3x3
4. When x = 5: y = 3 x 125 = 375

y ∝ square root of x

Worked Example 4

y is directly proportional to the square root of x. When x = 16, y = 12. Find y when x = 25.

1. y = k times the square root of x
2. 12 = k x 4, so k = 3
3. Equation: y = 3 times the square root of x
4. When x = 25: y = 3 x 5 = 15

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Inverse Proportion

If y is inversely proportional to x, we write:

$y \propto \frac{1}{x}$y∝x1​, which means $y = \frac{k}{x}$y=xk​

$y \propto \frac{1}{x}$y∝x1​ (y is inversely proportional to x)

Worked Example 5

y is inversely proportional to x. When x = 3, y = 12. Find y when x = 9.

1. $y = \frac{k}{x}$y=xk​
2. 12 = k/3, so k = 36
3. Equation: $y = \frac{36}{x}$y=x36​
4. When x = 9: y = 36/9 = 4

$y \propto \frac{1}{x^2}$y∝x21​ (y is inversely proportional to x squared)

Worked Example 6

y is inversely proportional to x^2. When x = 2, y = 5. Find y when x = 4.

1. $y = \frac{k}{x^2}$y=x2k​
2. 5 = k/4, so k = 20
3. Equation: $y = \frac{20}{x^2}$y=x220​
4. When x = 4: y = 20/16 = 1.25

$y \propto \frac{1}{x^3}$y∝x31​ (y is inversely proportional to x cubed)

Worked Example 7

y is inversely proportional to x^3. When x = 2, y = 10. Find y when x = 5.

1. $y = \frac{k}{x^3}$y=x3k​
2. 10 = k/8, so k = 80
3. Equation: $y = \frac{80}{x^3}$y=x380​
4. When x = 5: y = 80/125 = 0.64

Edexcel Exam Tip: Always write down the proportionality statement (e.g. $y \propto x^2$y∝x2), convert it to an equation with k ($y = kx^2$y=kx2), then find k. This structured approach earns all the method marks.

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Finding Unknown Values

Worked Example 8

y is directly proportional to x^2. When x = 6, y = 108.

(a) Find x when y = 48.

1. $y = kx^2$y=kx2
2. 108 = k x 36, so k = 3
3. $y = 3x^2$y=3x2
4. 48 = 3x^2, so x^2 = 16, so x = 4 (taking the positive root)

(b) Find y when x = 10.

• y = 3 x 100 = 300

Worked Example 9

y is inversely proportional to x. When x = 5, y = 8. Find x when y = 2.

1. $y = \frac{k}{x}$y=xk​
2. 8 = k/5, so k = 40
3. $y = \frac{40}{x}$y=x40​
4. 2 = 40/x, so x = 40/2 = 20

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Graphical Representations

Direct Proportion Graphs

Relationship	Graph of y against x
$y \propto x$y∝x	Straight line through origin
$y \propto x^2$y∝x2	Parabola through origin
$y \propto x^3$y∝x3	Steeper curve through origin
y ∝ square root of x	Curve through origin, rising then flattening

Inverse Proportion Graphs

Relationship	Graph of y against x
$y \propto \frac{1}{x}$y∝x1​	Curve approaching both axes (rectangular hyperbola)
$y \propto \frac{1}{x^2}$y∝x21​	Steeper curve approaching both axes

Key Test for Direct Proportion

If y is directly proportional to x^2, then a graph of y against x^2 will be a straight line through the origin.

Similarly, if $y \propto \frac{1}{x}$y∝x1​, then a graph of y against 1/x will be a straight line through the origin.

Worked Example 10

The table shows values of x and y.

x	2	3	5	8
y	12	27	75	192

Show that y is directly proportional to x^2.

• Test: y/x^2 should be constant.
• 12/4 = 3; 27/9 = 3; 75/25 = 3; 192/64 = 3.
• Since y/x^2 = 3 for all values, $y \propto x^2$y∝x2 with k = 3.

Worked Example 11

Which of the following equations represents inverse proportion?

(a) $y = 3x$y=3x (b) $y = \frac{12}{x}$y=x12​ (c) y = x^2 + 1 (d) $y = 5x$y=5x - 2

Answer: (b) $y = \frac{12}{x}$y=x12​ — this is in the form $y = \frac{k}{x}$y=xk​.

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Setting Up Proportion Equations from Context

Worked Example 12

The time T to cook a joint of meat is directly proportional to the weight W of the joint. A 2 kg joint takes 90 minutes. How long does a 3.5 kg joint take?

1. $T \propto W$T∝W, so T = kW
2. 90 = k x 2, so k = 45
3. T = 45W
4. When W = 3.5: T = 45 x 3.5 = 157.5 minutes (or 2 hours 37.5 minutes)

Worked Example 13

The intensity of light I is inversely proportional to the square of the distance d from the source. At a distance of 2 metres, the intensity is 100 units. Find the intensity at 5 metres.

1. $I \propto \frac{1}{d^2}$I∝d21​, so $I = \frac{k}{d^2}$I=d2k​
2. 100 = k/4, so k = 400
3. $I = \frac{400}{d^2}$I=d2400​
4. When d = 5: I = 400/25 = 16 units

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Practice Problems