EST I HS Math · Unit 2 · Data Analysis & Probability
Lesson 2.4 · Direct, Inverse & Joint Variation
On the paper: A handful of real questions per year — but every one uses the same three moves
Big idea. Turn the sentence into an equation with a constant k FIRST — y = kx (direct), y = k/x (inverse), or z = kxy (joint) — solve for k from the pair you're given, then plug in the new value. Never guess a shortcut ratio when a power is involved.
1Direct variation: y = kx 📈
🍼 Baby steps
- 1
'selling price is directly proportional to area' → price = k × area
October 2020. "Directly proportional to" always means y = kx.
- 2
168,275 = k × 127 → k = 1,325 $/sqm
Find k from the ONE pair you know.
- 3
price at 156 sqm = 1,325 × 156 = $206,700
Now use the SAME k for the new value.
- 4
'length directly proportional to weight' — 5 m weighs 2 kg → k = 0.4 kg/m
January 2023. 2 m weighs 0.4 × 2 = 0.8 kg.
Trap · a power changes everything
2Inverse variation: y = k/x 📉
🍼 Baby steps
- 1
'h varies inversely as k' → h = c/k (using c so it isn't confused with the variable k)
October 2021. h = 9 when k = 20 → c = h × k = 180.
- 2
h = 15 → new k = 180 ÷ 15 = 12 → k + 4 = 16
Inverse variation: the PRODUCT stays constant, not the ratio.
- 3
'a varies inversely as b': a = 2, b = 8 → constant = 16
ab = 16 always, for this relationship.
- 4
b = 6 → a = 16 ÷ 6 = 8/3 → 9a = 9 × 8/3 = 24
Find a first, then whatever the question actually asks for (here, 9a — read to the end).
3A real-world law: Ohm's law, I = U/R ⚡
🍼 Baby steps
- 1
current I is directly proportional to voltage U, and inversely proportional to resistance R
March 2022. The formula IS I = U ÷ R — read which letter is on top.
- 2
if U increases (R fixed), I increases — direct relationship
U is on top: more voltage, more current.
- 3
if R increases (U fixed), I decreases — inverse relationship
R is on the bottom: more resistance, less current.
🎛 Play with it · direct vs inverse
Pick direct or inverse, enter one known (x, y) pair to find k, then slide a new x and watch how y responds on the curve.
y = 5x → at x = 6, y = 30
Direct: x and y grow together — double one, double the other. Inverse: x and y trade off — double x, y is cut in half. Either way, find k FIRST from the known pair, then plug in the new value.
4Real EST questions, solved twice 📄
The selling price of an apartment is directly proportional to its area. If an apartment of 127 sqm is sold for $168,275, what would be the price of a 156 sqm apartment?
- A.$237,900
168,275 × 1.4142 — no real route to this number.
- B.$208,500
Rounds the constant to 1,337.5.
- C.$136,993
Divides instead of multiplying by the ratio of areas.
- D.$206,700
Safe way · ~25 s
- k = 168,275 ÷ 127 = 1,325.
- 156 × 1,325 = 206,700.
⏱ 90-second way · ~10 s
- 168,275 × (156 ÷ 127) = 206,700. Scale by the ratio of the SAME quantity.
Answer: D
If y varies directly as the cube of 2x, and y = 8 when x = 2, what is y when x is equal to 4?
- A.16
Treats the power as 1: doubles y for doubling x.
- B.32
Treats the power as 2 (a square, not a cube).
- C.64
- D.512
Forgets to divide out k, using y = (2x)³ directly.
Safe way · ~35 s
- y = k(2x)³. 8 = k(4)³ = 64k → k = 1/8.
- At x = 4: y = (1/8)(8)³ = 512/8 = 64.
⏱ 90-second way · ~15 s
- x doubled → (2x) doubled → y multiplies by 2³ = 8. 8 × 8 = 64.
Answer: C
If h varies inversely as k, and h = 9 when k = 20, what is the value of k + 4 when h = 15?
- A.12
Forgets to add 4 at the end.
- B.13.3
Divides 20 by 15 instead of using the constant product.
- C.16
- D.24
Uses h × 15 instead of solving h·k = 180 for the new k.
Safe way · ~30 s
- h·k = 9 × 20 = 180 (constant).
- h = 15 → k = 180 ÷ 15 = 12 → k + 4 = 16.
⏱ 90-second way · ~12 s
- 9 × 20 = 180. 180 ÷ 15 = 12. 12 + 4 = 16.
Answer: C
If the length of an object y is directly proportional to its weight x, and a 5-meter length weighs 2 kilograms, how many kilograms is a 2-meter length of the same object?
- A.0.4 kg
This is the constant k itself, not the weight of a 2 m length.
- B.0.8 kg
- C.2.5 kg
Inverts the ratio (5 ÷ 2 instead of 2 ÷ 5).
- D.5 kg
Copies the original length's number by mistake.
Safe way · ~20 s
- k = 2 ÷ 5 = 0.4 kg per meter.
- 2 m × 0.4 = 0.8 kg.
⏱ 90-second way · ~8 s
- 2 × (2 ÷ 5) = 0.8 kg. Scale by the ratio of lengths.
Answer: B
⏱ The 90-second lane
One question = 90 seconds. These are the moves that finish this topic in one line.
- 1"Directly proportional to" → y = kx. "Inversely proportional to" → y = k/x. "Jointly" → z = kxy.
- 2ALWAYS find k from the pair you're given first — only then plug in the new value.
- 3A power inside the variation (square, cube) means doubling x does NOT double y — it multiplies y by 2 raised to that power.
- 4Direct variation: the RATIO y/x stays constant. Inverse variation: the PRODUCT xy stays constant.
- 5In a real-world formula (Ohm's law, etc.), whatever is on TOP varies directly with the answer; whatever is on the BOTTOM varies inversely.
- 6Read to the end of the question — it may ask for k, or for x, or for something built from them (like "k + 4" or "9a").
Take with you
- ▸Direct variation (y = kx) and inverse variation (y = k/x) both start the same way: solve for k using the one pair you know.
- ▸A power on the variable changes the SCALE factor, not the method — 2× the input becomes 2ᵖ× the output for a variation of degree p.
- ▸Joint variation just multiplies more than one variable into the same k-equation.