Variation is the maths of “as one quantity changes, how does the other respond?” The whole method is: find the constant that links them, then scale. We finish the chapter with a few smart ways to evaluate polynomials for awkward numbers.
Find the constant, then scale
• Direct variation:y∝x means y=kx. Use the first pair to find k, then apply it. As x grows, y grows.
• Inverse variation:y∝x1 means the product xy=k stays fixed. Multiply one variable by n and the other is divided by n.
• Chained powers: treat ∝ like = and substitute. If x∝y2 and y∝z, then x∝z.
Smart polynomial values: re-shape before you compute. p3+3p2+3p=(p+1)3−1, and for close numbers x3+y3+z3−3xyz=21(x+y+z)[(x−y)2+(y−z)2+(z−x)2].
🔷Identity ExplorerPick an identity to see its proof
(a+b)² = a² + 2ab + b²
Expand as (a+b)×(a+b). The square area splits into 4 rectangles: a², ab, ab, b².
Numeric Verification
LHS: (3+2)² = 25
RHS: 3²+2·3·2+2² = 25
✓ LHS = RHS — identity verified!
If y∝x and y=15 when x=5, then k= (so y=3x).
📝Practice Questions
Q1If y varies directly as x and y = 12 when x = 3, find y when x = 7.
Q2P varies inversely as Q. If P = 10 when Q = 6, find P when Q = 15.
Q3If p = 49, the value of p(p² + 3p + 3) follows (p+1)³ − 1. The cube used is:
📚
Real exam questions — Variation & Polynomials
2 question types · 8 solved examples from real SSC papers
Variation is just “find the constant, then scale”. Polynomial-value questions reward re-shaping the expression into a clean cube before you compute.
How to solve this type
Direct: y∝x means y=kx — find k from the first pair, then use it. Inverse: y∝x1 means xy=k stays constant. Powers carry through: x∝y2 and y∝z chain into x∝z. Ratio shortcut for inverse: multiply one variable by n ⇒ divide the other by n.
If y∝x and y=15 when x=5, find y when x=12.
A24B36C60D48
P varies inversely as Q. P=15 when Q=4. Find P when Q=12.
