What is inverse variation?
Two amounts vary inversely when one goes up and the other goes down so that their product (the answer when you multiply them) stays the same.
x × y = k, or y = k / x, where k is a fixed number called the constant of variation (or constant of proportionality). k is never 0.
We also say "y is inversely proportional to x" and write y ∝ 1/x.
Check: double x → y halves. Make x three times bigger → y becomes one third. Not every "one up, one down" pair is inverse! For y = 10 − x, y goes down, but x × y is not fixed, so it is not inverse variation.
How to find k and use it
- Write y = k / x (or xy = k).
- Put in one known pair to find k: k = x × y.
- Write the rule with that k.
- Put in the new x (or y) to find the missing value.
Short cut for two pairs: x₁ × y₁ = x₂ × y₂.
Test a table: multiply each pair. If every product is the same, it is inverse variation.
The graph of y = k/x
The graph is a hyperbola: two smooth curved branches.
- For k > 0 the branches are in the 1st and 3rd quadrants. For k < 0 they are in the 2nd and 4th.
- The curve never touches the x-axis or y-axis. These axes are its asymptotes (lines the curve gets closer to forever).
- x = 0 is not allowed (you cannot divide by 0), and y is never 0.
- For k > 0, as x gets bigger, y gets smaller (in each branch).
- The graph is symmetric about the lines y = x and y = −x.
- A bigger k pushes the curve further from the origin.
Compare with y = x² and y = √x
y = x² is a U-shaped parabola, and y = √x is a half-curve that starts at (0, 0) and rises slowly (only for x ≥ 0). Both rise for positive x. y = k/x falls. Direct variation y = kx is a straight line through the origin.
Real problems
Ask: "If I double one amount, does the other halve?" Common inverse pairs:
- speed and time for a fixed distance
- number of workers and days for a fixed job (same speed of work)
- number of people and share of a fixed amount
- pressure and volume of a gas at fixed temperature (Boyle's law)
- number of pipes and time to fill a tank
Always check units and that the fixed thing really stays fixed.
Try it
Take 24 coins, buttons or blocks. Make rectangles: 1 × 24, 2 × 12, 3 × 8, 4 × 6. Write width and height in a table and multiply. Then plot the pairs on squared paper and join them with a smooth curve. Predict the height for width 5 before you build it (24 ÷ 5 = 4.8, so it cannot be whole!). Check in the free-play step.
Key formulas and definitions
- xy = k
- y = k / x (x ≠ 0)
- y ∝ 1/x
- x₁y₁ = x₂y₂
- Direct: y = kx; Inverse: y = k/x
Worked examples
1. y varies inversely as x, and y = 6 when x = 4. Find y when x = 8.
Step 1: k = xy = 4 × 6 = 24. Step 2: y = 24/x. Step 3: y = 24/8 = 3.
2. Is this table inverse variation? x: 2, 3, 5; y: 15, 10, 6.
Products: 2 × 15 = 30, 3 × 10 = 30, 5 × 6 = 30. All equal, so yes, k = 30 and y = 30/x.
3. A car takes 3 h at 60 km/h. How long at 90 km/h for the same trip?
Speed × time = 60 × 3 = 180 (the distance). Time = 180 ÷ 90 = 2 h.
4. 8 workers build a wall in 15 days. How many days for 12 workers?
workers × days = 8 × 15 = 120. Days = 120 ÷ 12 = 10 days.
5. y ∝ 1/x and y = −5 when x = 2. Write the rule and say which quadrants the graph lies in.
k = 2 × (−5) = −10, so y = −10/x. k < 0, so the branches are in the 2nd and 4th quadrants.
6. The pressure of a gas is 100 kPa when its volume is 6 L. Temperature stays the same. Find the pressure at 4 L.
P₁V₁ = P₂V₂: 100 × 6 = P₂ × 4, so P₂ = 600 ÷ 4 = 150 kPa.
Common mistakes
- Dividing instead of multiplying to find k. For inverse variation k = x × y, not y ÷ x.
- Thinking any "one up, one down" pair is inverse. Check that the product is fixed.
- Letting the graph touch or cross the axes. y = k/x never meets x = 0 or y = 0.
- Forgetting that a negative k puts the curve in the 2nd and 4th quadrants.