What you'll learn
Variation describes how one quantity changes in relation to another, and it is an important algebra topic in the CSEC Mathematics syllabus. You will meet direct variation (as one quantity grows, the other grows proportionally), inverse variation (as one grows, the other shrinks), and joint variation (a quantity depends on two or more others). In this guide you will learn to write variation statements as equations using a constant of proportionality k, to find k from given data, and to use the completed formula to solve real problems — from fuel costs and wages to the time a job takes with more workers. Variation appears in Paper 1 and Paper 2 and rewards a clear, step-by-step method.
Key terms and definitions
Variation — a relationship in which quantities change together according to a fixed rule.
Direct variation — y varies directly as x, written y ∝ x, meaning y = kx.
Inverse variation — y varies inversely as x, written y ∝ 1/x, meaning y = k ÷ x.
Joint variation — y depends on the product (or quotient) of two or more variables, e.g. y = kxz.
Constant of proportionality (k) — the fixed number that turns a proportion sign into an equation.
Proportional — increasing or decreasing at a constant ratio.
Core concepts
Direct variation
If y varies directly as x, then y ∝ x, which becomes the equation y = kx for some constant k. Doubling x doubles y; tripling x triples y. The graph of y against x is a straight line through the origin with gradient k. A variable can vary directly as a power too: "y varies directly as the square of x" means y = kx².
Inverse variation
If y varies inversely as x, then y ∝ 1/x, which becomes y = k ÷ x. Now increasing x decreases y in the same ratio: doubling x halves y. A classic example is the time to complete a fixed job: more workers means less time, so time varies inversely as the number of workers. The graph of y against x is a curve (a hyperbola), never touching the axes.
Joint variation
Joint variation involves more than two quantities. "z varies jointly as x and y" means z = kxy. A quantity can vary directly as one variable and inversely as another at the same time: "y varies directly as a and inversely as b" gives y = ka ÷ b. These combined statements are common in Paper 2.
The three-step method
Every variation problem follows the same reliable steps. (1) Write the proportionality as an equation with k. (2) Substitute the given pair of values to find k. (3) Rewrite the formula with k filled in, then use it to answer the question. Keeping these steps separate prevents confusion.
Recognising the type
Read the wording carefully. "Varies as", "directly proportional to", or "in proportion to" → direct variation. "Varies inversely as", "inversely proportional to" → inverse variation. "Varies jointly as" or two linked variables → joint variation. The phrase tells you exactly which equation to write.
Worked examples
Example 1: Direct variation (Paper 2 style)
The cost C of fuel varies directly as the number of litres L. When L = 20, C = $96. Find C when L = 35.
Write C = kL. Substitute: 96 = k × 20, so k = 96 ÷ 20 = 4.8. The formula is C = 4.8L. When L = 35, C = 4.8 × 35 = $168.
Example 2: Inverse variation (Paper 2 style)
The time t taken to build a wall varies inversely as the number of workers n. With 6 workers it takes 10 days. How long with 15 workers?
Write t = k ÷ n. Substitute: 10 = k ÷ 6, so k = 60. The formula is t = 60 ÷ n. With n = 15, t = 60 ÷ 15 = 4 days.
Example 3: Joint variation (Paper 2 style)
y varies directly as x and inversely as the square of z. When x = 12 and z = 2, y = 9. Find y when x = 10 and z = 5.
Write y = kx ÷ z². Substitute: 9 = k × 12 ÷ 2² = 12k ÷ 4 = 3k, so k = 3. The formula is y = 3x ÷ z². When x = 10 and z = 5, y = 3 × 10 ÷ 25 = 30 ÷ 25 = 1.2.
Common mistakes and how to avoid them
Confusing direct and inverse. "Inversely" means divide by the variable (y = k ÷ x), so the result gets smaller as x grows. Re-read the key word before writing the equation.
Skipping the constant k. A proportion sign (∝) is not an equation. You must introduce k and find its value from the given data.
Forgetting a power. "Varies as the square of x" means kx², and "inversely as the square of z" means k ÷ z². Include the power exactly as stated.
Reusing k incorrectly. Find k once from the given pair, then keep it fixed for the rest of the problem.
Not rewriting the full formula. After finding k, write the complete equation before substituting the new values; this keeps the work organised.
Exam technique for Variation
Translate the words into an equation immediately. Direct → y = kx; inverse → y = k ÷ x; joint → y = kxy or combinations.
Use the three-step method every time. Equation with k → find k → rewrite and solve.
Show the value of k clearly. It is usually worth a mark and is needed for the rest of the question.
Check the direction of change. In direct variation answers grow together; in inverse variation, as one grows the other shrinks. Use this to sanity-check.
Include units where the context (cost, time, distance) requires them.
Quick revision summary
Variation describes how quantities change together. Direct variation (y ∝ x) becomes y = kx — both grow in the same ratio and the graph is a straight line through the origin. Inverse variation (y ∝ 1/x) becomes y = k ÷ x — as one grows the other shrinks, like time against number of workers. Joint variation links three or more quantities, such as z = kxy, or combined direct-and-inverse forms like y = ka ÷ b. Variables can also vary as a power, e.g. y = kx². Solve every problem with the same three steps: write the proportionality as an equation containing the constant k, substitute the given values to find k, then rewrite the full formula and use it to answer. Read the wording carefully to pick the correct type, never drop the constant k or a stated power, and check that your answer changes in the expected direction.