Variation · Form 5
Inverse variation, explained
Inverse variation means that as one quantity grows, the other shrinks in the same proportion, so their product stays fixed. We write y ∝ 1/x, which becomes y = k/x for a constant k.
What inverse variation says
If y varies inversely as x, then doubling x halves y and tripling x cuts y to a third, the product xy always equals the same constant k. A classic picture is a fixed journey: at a steady faster speed you finish in less time, and speed × time stays equal to the distance.
So for a 60 km trip, 60 km/h takes 1 hour and 120 km/h takes ½ hour, yet speed × time = 60 throughout.
How to recognise it in a question
Look for phrases like 'varies inversely as', 'inversely proportional to', or a table where the product of each pair is the same number. The graph is a smooth curve (a hyperbola) that falls steeply then levels off, never touching either axis, because y = k/x is undefined when x = 0.
It can also involve a power, such as y ∝ 1/x², where x²y is the constant.
The usual misconception
A relationship where y simply decreases as x increases is not automatically inverse variation. In y = 10 − x, y falls as x rises, yet the product xy keeps changing, so it is not inverse variation.
The real test is a constant product xy = k; students also mix this up with direct variation, so always check whether the quantities move the same way (direct) or opposite ways (inverse).
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
How do I recognize inverse variation from a table of values?
Check whether the product xy stays the same for every pair in the table, that constant is k. As x gets larger, y should get smaller (or vice versa) so their product doesn't change.
If xy is constant across all pairs, it's inverse variation, y = k/x.
How do I find k and use y = k/x to solve exam problems?
Substitute a known pair of x and y into y = k/x and solve for k. Rewrite the equation with that k value, then substitute any given x or y to find the missing quantity.
Typical questions give one pair to find k, then ask you to find y when x changes.
What mistake do students often make with inverse variation?
Students often confuse y = k/x with y = k/x², which is a different relationship, always check the exact power stated in the question. Another common slip is writing y = kx instead of y = k/x, mixing up direct and inverse variation.
Read the wording carefully before setting up the equation.