Combined variation puts direct, inverse and joint relationships into one sentence. Read it phrase by phrase, place each quantity on top or below, and add a single k.
This lesson follows working with joint variation and closes the variation section.
How do I translate the wording?
Use this table to turn each piece of wording into a position in the equation.
| Wording | Where it goes | Example |
|---|---|---|
| varies directly as x | x on top | y = kx |
| varies inversely as z | z below | y = k/z |
| varies as the square of x | x² on top | y = kx² |
| varies inversely as the square root of z | √z below | y = k/√z |
| jointly as x and w | both on top | y = kxw |
A sentence joins these with “and”. Write all the top pieces together, all the below pieces together, then add k.
Worked example: a square and a square root
y varies directly as the square of x and inversely as the square root of z. y = 18 when x = 3 and z = 16. Find y when x = 4 and z = 4.
- Write y = kx²/√z.
- Substitute 18 = k × 9 ÷ 4, so k = 18 × 4 ÷ 9 = 8.
- The equation is y = 8x²/√z.
- For x = 4 and z = 4: y = 8 × 16 ÷ 2 = 64.
Check the direction of change: x rose, which raises y, and z fell, which also raises y. Going from 18 to 64 is consistent.
The mistake that costs marks
The common slip is to put every quantity on top. A student reads “inversely as the square root of z” and still writes y = kx²√z.
| Step | Wrong | Right |
|---|---|---|
| Equation | y = kx²√z | y = kx²/√z |
| Find k | 18 = k × 9 × 4, so k = 0.5 | 18 = k × 9 ÷ 4, so k = 8 |
| y at x = 4, z = 4 | 0.5 × 16 × 2 = 16 | 8 × 16 ÷ 2 = 64 |
The word “inversely” attaches to z only. The phrase before “and” stays on top.
Check yourself
M varies jointly as p and q², and inversely as r. M = 10 when p = 2, q = 5 and r = 5. Find M when p = 4, q = 3 and r = 6.
Answer
Write M = kpq²/r. Substitute 10 = k × 2 × 25 ÷ 5 = 10k, so k = 1.
For p = 4, q = 3 and r = 6: M = 4 × 9 ÷ 6 = 6.
What to study next
Mixed questions are where this skill is tested, so move to the variation practice set. Keep a record of each setup error in the mistake log and paper-error review so you can see which phrase keeps causing slips.
For a teacher to go through these with you, see online one-to-one Mathematics tuition.