Mathematics · Lesson 6

Variation (direct, inverse, joint, partial)

JAMBWAECNECO

What the syllabus asks

JAMB: Algebra: variation (direct, inverse, joint, partial), percentage increase and decrease in variation

WAEC: Algebraic processes: variation: direct, inverse, joint and partial variation; application to real-life problems

NECO: Algebraic processes: variation: direct, inverse, joint and partial variation; application to real-life problems

What you will learn

Translate statements of variation into equations with k.
Solve problems on direct and inverse variation, including powers and roots.
Solve problems on joint variation.
Solve partial variation problems using simultaneous equations.
Find the percentage change in one quantity caused by a change in another.
Recognise variation graphs and apply variation in science and daily life.

Key points

Replace ∝ with = k×; find k from given values; rewrite and use.
Direct: y = kx (y/x constant). Inverse: y = k/x (xy constant).
Powers and roots: y = kx², y = kx³, y = k√x, y = k/x².
Joint: multiply the variables; "inversely" goes underneath.
Partial: y = a + bx needs two pairs of values and simultaneous equations.
Percentage change: use multipliers, e.g. 1.1² = 1.21 gives 21%.
Graphs: direct is a line through O; partial is a line with intercept; inverse is a curve.
Check answers: does y rise or fall sensibly as x changes?

Start here: What variation means

In everyday life, many quantities depend on each other. If you buy more loaves of bread at ₦1,200 each, you pay more. If more people share the cost of a bus, each person pays less. If a car travels faster, the journey takes less time. Variation is the mathematics of these relationships.

The symbol ∝ means "varies as" or "is proportional to". Every variation statement can be turned into an equation by replacing ∝ with "= k ×", where k is a fixed number called the constant of variation.

y ∝ x ⟹ y = kx

The three-step method works for every variation problem:

Write the relationship as an equation with k.
Substitute the given pair of values to find k.
Write the equation with the value of k, then use it to answer the question.

Worked example 6.1

The cost C of bread varies directly as the number of loaves n. 5 loaves cost ₦6,000. Find the cost of 8 loaves.

C = kn.
6000 = 5k, so k = 1200 (the price of one loaf).
C = 1200n; for n = 8, C = ₦9,600.

₦9,600

Notice that k had a real meaning: the cost of one loaf. In many problems, finding what k represents helps you check that your answer is sensible.

⚠️ Common mistake: skipping step 3 and using the wrong k. Always write the final equation, e.g. C = 1200n, before substituting.

Try it yourself. y ∝ x and y = 12 when x = 4. Find y when x = 7.

Show solution

k = 3; y = 3 × 7 = 21.

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Practice questions

1. y ∝ x, y = 15 when x = 3. Find y when x = 11.

Show worked answer

k = 5; y = 55.

2. y ∝ 1/x, y = 8 when x = 3. Find x when y = 6.

Show worked answer

k = 24; x = 4.

3. P ∝ Q², P = 20 when Q = 2. Find P when Q = 3.

Show worked answer

k = 5; P = 45.

4. z varies directly as x and inversely as y; z = 6 when x = 9, y = 3. Find z when x = 12, y = 8.

Show worked answer

k = 2; z = 2 × 12/8 = 3.

5. C is partly constant and partly varies as n. C = 1,400 when n = 10 and C = 2,000 when n = 25. Find C when n = 40.

Show worked answer

b = 40, a = 1000; C = 2,600.

Key words

Variation: A relationship in which one quantity changes as another changes.
Constant of variation (k): The fixed number linking varying quantities.
Direct variation: y = kx: both increase together in the same ratio.
Inverse variation: y = k/x: as one increases, the other decreases in the same ratio.
Joint variation: One quantity varies as the product of two or more others.
Partial variation: A quantity made of a fixed part plus a part that varies.
Proportional (∝): The symbol meaning "varies as".
Square law: y ∝ x²: doubling x multiplies y by 4.
Inverse square law: y ∝ 1/x²: doubling x divides y by 4.
Multiplier: The factor by which a quantity is multiplied, e.g. 1.2 for a 20% rise.

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