Variation · Form 5
Variation: Revision Notes
A tight revision summary of Variation for SPM Mathematics, the key ideas, the formulae the exam gives you, and what to focus on before the paper.
The big idea
This chapter formalises the everyday idea that things change together. In direct variation one quantity rises as another rises; in inverse variation one rises as another falls; combined variation mixes several relationships.
You learn to form the equation, find the constant, and solve for an unknown.
Key ideas to revise
- Direct variation. y ∝ x means y = kx. Double x and y doubles. Find k from one pair of values, then use it everywhere.
- Inverse variation. y ∝ 1/x means y = k/x. As x grows, y shrinks, the shape is a curve, not a line.
- Finding the constant k. Every variation problem hinges on finding k first from the given values; everything else follows.
Work each key idea through with numbers
- Direct variation: y ∝ x becomes y = kx. If y = 15 when x = 3, then k = 15 ÷ 3 = 5, so y = 5x. Predict y when x = 7: y = 5 × 7 = 35.
- Inverse variation: y ∝ 1/x becomes y = k/x. If y = 6 when x = 4, then k = 6 × 4 = 24, so y = 24/x. Predict y when x = 8: y = 24 ÷ 8 = 3.
- Finding k with a power: y ∝ x² becomes y = kx². If y = 50 when x = 5, then k = 50 ÷ 25 = 2, so y = 2x². Predict y when x = 3: y = 2 × 9 = 18.
A pre-paper checklist for variation
- Re-derive, don't memorise: turn every 'varies as' into ∝, then swap ∝ for = k. Direct multiplies (y = kx), inverse divides (y = k/x), a power stays on its own variable (y = kx²).
- What the exam gives you: a relationship in words plus ONE complete pair of values to pin down k, then a second incomplete set to solve.
- The one habit that saves marks: find k first and box it, write the full equation, then substitute, never jump straight from the words to the final number.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
In what order should I work a variation question under time pressure?
Write the words as a proportionality (∝), replace ∝ with = k to form the equation, substitute the complete pair to find k, box k, write the full equation, then substitute the second set. Following the same five steps every time turns a wordy question into a mechanical routine and stops direct-inverse mix-ups.
Do joint and combined variation need their own formulae to memorise?
No. Build them from the proportionality every time.
'Jointly as x and z' becomes y = kxz; 'directly as x and inversely as z' becomes y = kx/z. If you learn the ∝ statement instead of a list of formulae, one method covers direct, inverse, joint and combined variation without confusion.
How can I check my value of k is correct before moving on?
Substitute the original pair back into your equation. If y = 5x and the given pair was y = 15, x = 3, check that 5 × 3 = 15.
When it matches, k is safe to reuse; when it does not, you have caught the error before it spreads into parts (b) and (c).