If z varies jointly as x and y, write z = kxy. One constant k links z to the product of the other two quantities.
This lesson follows solving inverse variation problems and belongs to the variation section.
How is joint variation different?
In direct variation, one quantity drives z. In joint variation, two or more drive it together, and they multiply. Use the same method: write the equation with k, find k from one set of values, then solve.
Worked example: the cost of carpet
The cost C of a carpet varies jointly as its length l and its width w. A carpet with l = 6 m and w = 4 m costs RM360. Find the cost of a carpet with l = 8 m and w = 5 m.
- Write C = klw.
- Substitute 360 = k × 6 × 4 = 24k, so k = 15.
- The equation is C = 15lw. For l = 8 and w = 5, C = 15 × 40 = RM600.
Check: the area rose from 24 m² to 40 m², a factor of 40 ÷ 24 = 1.667, and 360 × 1.667 = 600.
The mistake that costs marks
The common slip is to add the quantities. A student reads “jointly” as “together”, writes C = k(l + w) and finds k.
| Step | Wrong | Right |
|---|---|---|
| Equation | C = k(l + w) | C = klw |
| Find k | 360 = k × 10, so k = 36 | 360 = k × 24, so k = 15 |
| Cost at l = 8, w = 5 | 36 × 13 = RM468 | 15 × 40 = RM600 |
A quick test exposes the wrong form. If the width is doubled with the length unchanged, the cost should double. Under the sum form, it would not.
Check yourself
V varies jointly as r² and h. V = 150 when r = 5 and h = 6. Find V when r = 4 and h = 9.
Answer
Write V = kr²h. Substitute 150 = k × 25 × 6 = 150k, so k = 1.
For r = 4 and h = 9: V = 1 × 16 × 9 = 144.
Check with ratios: r² changes by 16 ÷ 25 and h by 9 ÷ 6, so V = 150 × 0.64 × 1.5 = 144.
What to study next
Finish the set with translating combined variation into equations, where direct, inverse and joint appear in one sentence. Then test yourself in the variation practice set. If the arithmetic of solving for k is the weak spot, try the algebra step repair trainer.
For a teacher to go through these with you, see online one-to-one Mathematics tuition.