Algebra and Graphs: Direct and inverse proportion — algebraic and graphical

What you'll learn

Direct and inverse proportion form a fundamental component of the CIE IGCSE Mathematics syllabus, appearing regularly in both Paper 2 and Paper 4 examinations. This topic combines algebraic manipulation with graphical interpretation, requiring you to recognise proportional relationships, form equations using constants of proportionality, and sketch or analyse characteristic graphs. Mastery of these concepts enables you to solve real-world problems involving speed, density, physics relationships, and economic scenarios.

Key terms and definitions

Direct proportion — a relationship between two variables where one is a constant multiple of the other; when one variable doubles, the other doubles; expressed as y ∝ x or y = kx where k is the constant of proportionality.

Inverse proportion — a relationship where one variable increases as the other decreases in such a way that their product remains constant; when one variable doubles, the other halves; expressed as y ∝ 1/x or y = k/x.

Constant of proportionality (k) — the fixed multiplier that connects two proportional variables; found by substituting known values into the proportion equation.

Proportionality symbol (∝) — means "is proportional to" and indicates a relationship exists but does not specify the constant; must be converted to an equation with = and k.

Joint variation — when a variable is proportional to multiple other variables, such as y ∝ xz or y ∝ x/z.

Square proportion — relationships involving squares or square roots, such as y ∝ x² or y ∝ √x.

Core concepts

Recognising direct proportion

Two quantities are in direct proportion when their ratio remains constant. Algebraically, if y is directly proportional to x:

• The relationship is written as y ∝ x
• This converts to the equation y = kx where k ≠ 0
• The graph is a straight line passing through the origin
• k represents the gradient of this line

Real examination contexts include:

• Cost proportional to quantity purchased
• Distance proportional to time (at constant speed)
• Extension of a spring proportional to applied force (Hooke's Law)
• Currency conversion rates

When identifying direct proportion from data, check whether y/x gives the same value for all pairs. If the quotient is constant, direct proportion exists.

Working with direct proportion algebraically

Step-by-step method for direct proportion problems:

1. Write the proportionality statement: y ∝ x
2. Convert to an equation: y = kx
3. Substitute the given values to find k
4. Write the complete equation with the numerical value of k
5. Use this equation to find unknown values

For square relationships where y ∝ x²:

• The equation becomes y = kx²
• The graph is a parabola passing through the origin
• Common in physics (energy proportional to velocity squared)

For square root relationships where y ∝ √x:

• The equation becomes y = k√x
• The graph is half a parabola on its side
• Appears in pendulum problems (period proportional to square root of length)

Recognising inverse proportion

Two quantities are in inverse proportion when their product remains constant. Algebraically, if y is inversely proportional to x:

• The relationship is written as y ∝ 1/x
• This converts to the equation y = k/x or xy = k
• The graph is a rectangular hyperbola with asymptotes along both axes
• As x increases, y decreases (and vice versa)

Typical examination contexts include:

• Time inversely proportional to speed (for fixed distance)
• Number of workers inversely proportional to time to complete a task
• Pressure inversely proportional to volume (Boyle's Law, at constant temperature)
• Brightness inversely proportional to distance squared

When checking for inverse proportion from data, verify whether xy produces the same value for all pairs. A constant product confirms inverse proportion.

Working with inverse proportion algebraically

Step-by-step method for inverse proportion problems:

1. Write the proportionality statement: y ∝ 1/x
2. Convert to an equation: y = k/x
3. Substitute given values to find k
4. Write the complete equation: y = k/x
5. Use this equation to find unknown values

For inverse square relationships where y ∝ 1/x²:

• The equation becomes y = k/x²
• The graph falls more steeply than simple inverse proportion
• Newton's law of gravitation and light intensity follow this pattern

Graphical characteristics

Direct proportion graphs (y = kx):

• Straight line through the origin (0, 0)
• Positive gradient when k > 0
• Negative gradient when k < 0
• The gradient equals k
• Extends infinitely in both directions

Inverse proportion graphs (y = k/x):

• Rectangular hyperbola shape
• Two separate branches in opposite quadrants
• Never touches the axes (asymptotic behaviour)
• When k > 0: branch in quadrant I (both positive) and quadrant III (both negative)
• When k < 0: branch in quadrants II and IV
• Symmetrical about the line y = x when k > 0

Distinguishing graphs in examinations:

• Direct proportion: linear, passes through origin
• y ∝ x²: curved upward, passes through origin
• Inverse proportion: hyperbola, approaches but never touches axes
• y ∝ 1/x²: steeper hyperbola, approaches axes faster

Joint and compound proportions

Some CIE IGCSE questions involve joint variation where one variable depends on multiple others:

When y ∝ xz:

• y = kxz
• y increases if either x or z increases
• Example: Area of rectangle proportional to length and width

When y ∝ x/z:

• y = kx/z
• y is directly proportional to x and inversely proportional to z
• Example: Speed is proportional to distance and inversely proportional to time

When y ∝ x²/z:

• y = kx²/z
• Combines square and inverse relationships
• Example: Kinetic energy proportional to mass and velocity squared

The method remains the same: state the relationship, convert to an equation with k, find k using given values, then solve for unknowns.

Worked examples

Example 1: Direct proportion with squares

Question: The distance d metres that a ball rolls down a slope is directly proportional to the square of the time t seconds. When t = 4, d = 32.

(a) Find an equation connecting d and t. [3 marks] (b) Calculate d when t = 7. [2 marks] (c) Calculate t when d = 50. [2 marks]

Solution:

(a) d ∝ t² [stating the relationship]

d = kt² [converting to equation with k]

Substitute d = 32 when t = 4:

32 = k(4²)

32 = 16k

k = 2

Therefore: d = 2t² [3 marks: 1 for d = kt², 1 for substitution, 1 for correct equation]

(b) When t = 7:

d = 2(7²) = 2(49) = 98 metres [2 marks: 1 for substitution, 1 for answer]

(c) When d = 50:

50 = 2t²

t² = 25

t = 5 (taking positive root as time is positive)

t = 5 seconds [2 marks: 1 for method, 1 for answer]

Example 2: Inverse proportion

Question: The time T hours taken to paint a fence is inversely proportional to the number of people P working on it. When 3 people work together, the fence takes 8 hours to paint.

(a) Express T in terms of P. [3 marks] (b) How long would 5 people take? [2 marks] (c) How many people are needed to complete the job in 4 hours? [2 marks]

Solution:

(a) T ∝ 1/P [stating inverse proportion]

T = k/P [equation form]

When P = 3, T = 8:

8 = k/3

k = 24

Therefore: T = 24/P [3 marks]

(b) When P = 5:

T = 24/5 = 4.8 hours [2 marks]

(c) When T = 4:

4 = 24/P

4P = 24

P = 6

6 people needed [2 marks]

Example 3: Identifying proportion from a graph

Question: The graph shows the relationship between variables x and y.

[Graph shows a curve passing through origin, increasing at a decreasing rate]

(a) State whether y is proportional to x, x², √x, or 1/x. [1 mark] (b) The point (4, 6) lies on the curve. Find the equation connecting y and x. [3 marks]

Solution:

(a) The curve passes through the origin and increases at a decreasing rate (concave down).

This indicates: y ∝ √x [1 mark]

(b) y = k√x

When x = 4, y = 6:

6 = k√4

6 = 2k

k = 3

Equation: y = 3√x [3 marks]

Common mistakes and how to avoid them

• Forgetting to include the constant k — writing y = x instead of y = kx when converting y ∝ x to an equation. Always include k immediately when you replace the proportionality symbol with an equals sign.

• Confusing direct and inverse proportion — identifying a relationship as direct proportion when one variable increases while the other decreases. Check: if both increase together (or both decrease), it's direct; if one increases while the other decreases, it's inverse.

• Assuming k = 1 — using the equation y = x or y = 1/x without calculating k from given values. You must always substitute known values to find the specific value of k for that problem.

• Incorrect manipulation of inverse proportion — writing y = x/k instead of y = k/x for inverse proportion. Remember: in inverse proportion, the constant k is in the numerator, and the variable is in the denominator.

• Misidentifying graphs — thinking any curve is inverse proportion. Direct proportion with squares (y = kx²) curves upward through the origin; inverse proportion (y = k/x) never touches the axes and has a hyperbola shape.

• Not taking the positive root — in contexts like time or distance, giving negative values when solving equations involving squares. Physical quantities like time, length, and speed are positive, so reject negative solutions.

Exam technique for Algebra and Graphs: Direct and inverse proportion — algebraic and graphical

• "Express," "find an equation," or "show that" — these command words require you to write the proportionality statement, convert it to an equation with k, substitute values, and solve for k. Show all three steps clearly for full marks (typically 3 marks: relationship, equation with k, finding k).

• Structure for proportion problems — follow this sequence every time: (1) state the proportionality using ∝, (2) write y = k... equation, (3) substitute to find k, (4) rewrite equation with numerical k, (5) use equation to answer the question. Missing step (1) or (2) commonly loses a method mark.

• Graph sketching marks — when asked to sketch a proportion graph, examiners award marks for: passing through origin (direct), correct general shape, showing asymptotic behaviour (inverse), and labelling axes. A rough sketch is acceptable but must show these key features.

• Units and context — always include units in your final answer when the question gives them (hours, metres, people, etc.). In inverse proportion with compound units (like speed = distance/time), track units carefully through the calculation.

Quick revision summary

Direct proportion (y ∝ x): ratio y/x is constant, equation y = kx, straight line through origin. Inverse proportion (y ∝ 1/x): product xy is constant, equation y = k/x, hyperbola graph with asymptotes on axes. Square proportions involve x² or √x. Always write the relationship, form equation y = k×..., substitute known values to find k, then solve. Direct proportion graphs are linear through origin; inverse proportion graphs curve towards axes but never touch. Joint variation combines multiple proportional relationships in one equation.