Variations
Set up direct, inverse, joint, and partial variation equations in seconds. Learn the fastest decimal shortcut for percentage change questions in Paper 2.
The Core Concept
How Variables Interact: Variations describe how one quantity changes when other quantities change. In Direct Variation, the variables grow or shrink together. In Inverse Variation, as one variable grows, the other shrinks. In Partial Variation, the relationship is divided into separate parts added together.
DSE Variations Cheat Sheet — 5 Key Topics
Scan this compact reference table to review every essential concept, step, and shortcut in under 10 seconds.
| Topic | Key Concept & Steps | Exam Shortcuts & Tips |
|---|---|---|
| 1. Direct Variation | y ∝ xⁿy = kxⁿ | • k ≠ 0 is a constant.• Tested with n = 1 (linear), n = 2 (square), or n = ½ (square root). |
| 2. Inverse Variation | y ∝ 1/xⁿy = k/xⁿ ⟺ yxⁿ = k | • k ≠ 0 is a constant.• Identify from phrasings like: "As x increases, y decreases." |
| 3. Joint Variation | y ∝ xᵐzⁿ• Direct & Inverse combined: y ∝ x/z²y = kxᵐzⁿy = kx/z² | Combines multiple variables. Requires only one constant of variation (k). |
| 4. Partial Variation | • Partly constant & partly varies directly:y = const + dir. as x• Linear-constant: y = k₁ + k₂x• Two variables: y = k₁x + k₂z² | Requires two constants (k₁ and k₂). You must set up and solve a system of simultaneous linear equations. |
| 5. Percentage Change | Compare ynew to yold after substituting changed values. | Express changes as decimals (e.g., +20% ⟹ 1.2x; −10% ⟹ 0.9z). |
💡 Exam Traps & Secrets
⚠️ The Non-Zero Constant (k ≠ 0) StatementIn Paper 1 Section A, whenever you introduce a constant of variation (such as k, k₁, or k₂), you must explicitly write that it is a non-zero constant. Failing to write "where k is a non-zero constant" can cost you a presentation mark!
⚠️ The Partial Variation Formula TrapWhen setting up partial variation equations, do not accidentally multiply the constants.
Incorrect:
Correct:
You must have two separate, independent constants (k₁ and k₂) to solve using simultaneous equations.
Incorrect:
y = k(1 + x)Correct:
y = k₁ + k₂xYou must have two separate, independent constants (k₁ and k₂) to solve using simultaneous equations.
⚠️ The Percentage Change Decimal Shortcut (Paper 2)When solving multiple-choice percentage change questions, avoid writing long algebra. Use the decimal substitution shortcut instead:
Problem: If
Shortcut: Replace x with
Multiplier =
Result: ynew ≈ 1.4815 yold. The percentage change is:
Problem: If
y = kx/z², and x increases by 20% while z decreases by 10%, find the percentage change in y.Shortcut: Replace x with
1.2 and z with 0.9:Multiplier =
1.2 / 0.9² = 1.2 / 0.81 ≈ 1.4815Result: ynew ≈ 1.4815 yold. The percentage change is:
(1.4815 − 1) × 100% = +48.15%✏️ Self-Test: Quick Interactive Practice
How to study: Click any question below to instantly load it on the coordinate grid and check your steps.
(Setting up inverse equations and solving for the constant k)y varies inversely as x. When
x = 4, y = 9. Find y when x = 12.(Joint variation equation solving)y varies jointly as x and the square of z. When
x = 2, z = 3, y = 36. Find y when x = 5 and z = 2.(Solving partial variation systems using simultaneous equations)y is partly constant and partly varies directly as x. When
x = 2, y = 11, and when x = 5, y = 20. Find y when x = 8.Variations Exercises: DSE-Style Practice
DSE-style practice sets from the SketchMath question bank, with solutions for every MC and long question.
Frequently Asked Questions
"Varies directly as x" → y = kx; "inversely as x" → y = k/x; "as the square of x" → y = kx²; "jointly as x and z" → y = kxz; "partly constant, partly as x" → y = k₁ + k₂x. The phrasing in the question directly tells you which equation form to use.
Substitute the new x (e.g. 1.2x₀) into the variation equation, simplify, and compare to original y₀. The percentage change is (new y − old y)/old y × 100%. For Paper 2 MC, use the decimal multiplier shortcut: replace each variable with its decimal factor (e.g., +20% → 1.2, −10% → 0.9) and compute the overall multiplier directly.
One condition per unknown constant. Direct/inverse variation has one constant (k), so one condition suffices. Partial variation y = k₁ + k₂x has two constants, requiring two conditions to form a system of simultaneous equations.