Direct & Inverse Variation
Some quantities grow together, others trade off. Direct variation (y = kx) means the ratio y/x is constant; inverse variation (y = k/x) means the product xy is constant. Learn to find the constant k and use it to predict.
1 Direct versus inverse
Direct variation: y = kx — as x grows, y grows in proportion, and the ratio y/x stays constant. Inverse variation: y = k/x — as x grows, y shrinks, and the product xy stays constant.
k is the constant of variation. Find it from one data point, then use it for every other point. Joint variation multiplies two inputs: y = kxz.
The test: given a table, compute y/x for each row — constant means direct. Compute xy — constant means inverse. Neither constant means neither.
2 Worked examples
Find k, then predict.
y varies directly with x, and y = 20 when x = 4. Find k.
k = y/x = 20/4 = 5, so the model is y = 5x.
With y = 5x from Example 1, find y when x = 7.
y = 5(7) = 35.
y varies inversely with x, and y = 8 when x = 3. Find k.
k = xy = 3 · 8 = 24, so the model is y = 24/x.
With y = 24/x, find y when x = 6.
y = 24/6 = 4 — bigger x, smaller y.
y varies jointly with x and z (y = kxz); y = 30 when x = 2, z = 3. Find k.
k = y/(xz) = 30/6 = 5.
3 Common mistakes
Three traps, each with the wrong version and the fix.
Wrong: “y varies inversely, y = 8 when x = 3” → k = 8/3. Right: inverse means the product is constant: k = 3 · 8 = 24.
Wrong: seeing (2, 12), (3, 8), (4, 6) and fitting y = kx. Right: check both: y/x varies (6, 2.67, 1.5) but xy = 24 always — it is inverse.
Wrong: “find y when x = 6” answered with k = 24. Right: k is the middle step — finish with y = k/x = 4.
4 Quick checks
Try these yourself, then reveal the answer.
5 Key points
Remember
- Direct: y = kx; the ratio y/x is constant.
- Inverse: y = k/x; the product xy is constant.
- Joint: y = kxz; solve k = y/(xz).
- Find k from the known point first, then predict.
- Test data with both y/x and xy before naming the variation.