The idea
y varies inversely with x when
xy = k or y = k/x
where k is the constant of variation. As x increases, y decreases, so their product stays the same.
If x doubles, y halves. If x triples, y becomes one third.
Examples
| Situation | Relationship |
|---|---|
| Time to drive 120 km at speed v | t = 120/v |
| Workers to finish a job | more workers, less time |
| Sharing a $60 prize among n people | each gets 60/n |
| Length and width of a rectangle with area 24 | l = 24/w |
Recognizing inverse variation in a table
Check whether x × y is constant.
| x | 1 | 2 | 3 | 4 | 6 |
|---|---|---|---|---|---|
| y | 12 | 6 | 4 | 3 | 2 |
| xy | 12 | 12 | 12 | 12 | 12 |
The product is always 12: y = 12/x.
The graph
The graph of y = k/x is a curve called a hyperbola. For positive x it:
- gets closer and closer to the x-axis as x grows, but never touches it;
- shoots up as x gets close to 0.
The axes are asymptotes: lines the curve approaches but never reaches. x can’t be 0, because division by zero is undefined.
Solving problems
Example: 4 painters can paint a house in 6 days. How long for 3 painters (same rate)?
- k = 4 × 6 = 24 painter-days
- Days = 24 ÷ 3 = 8 days
Example: At 80 km/h, a trip takes 3 hours. How long at 60 km/h?
- k = 80 × 3 = 240 km
- t = 240 ÷ 60 = 4 hours
Direct vs inverse
| Direct | Inverse |
|---|---|
| y = kx | y = k/x |
| y/x constant | xy constant |
| both increase together | one up, one down |
| straight line through origin | hyperbola |
Worked examples
Example 1: y varies inversely with x; y = 5 when x = 8. Find y when x = 10.
k = 40 → y = 4.
Example 2: Does x: 2, 5, 10; y: 25, 10, 5 show inverse variation?
Products: 50, 50, 50 → yes.
Example 3: A pizza is shared equally. With 4 people, each gets 3 slices. With 6 people?
k = 12 → 2 slices each.
Common mistakes
- Assuming any decreasing relationship is inverse. Check the product.
- Using a proportion (as for direct variation) and getting the wrong direction.
- Letting x = 0.
Try it yourself
- xy = 36. Find y when x = 9.
- y = 4 when x = 15. Write the equation.
- 6 pumps empty a pool in 10 h. How long for 5 pumps?
- Is y = 3/x + 1 an inverse variation?
Answers: 1) 4 2) y = 60/x 3) 12 h 4) No (the + 1 breaks it)
Quick recap
- Inverse variation: xy = k; as one goes up, the other goes down.
- Its graph is a hyperbola with the axes as asymptotes.
- Check that the product is constant.