If y varies inversely as x, write y = k/x, and then k is the product xy. Use one known pair to find k and substitute the new value.
This lesson follows solving direct variation problems and belongs to the variation section.
How do I spot an inverse variation?
Ask what happens when one quantity increases. If the other decreases in the same proportion, the product stays constant. More workers finish a job in less time, and a faster speed covers a fixed distance in less time.
Worked example: speed and time
On a fixed route, the speed v varies inversely as the time t. A car at 60 km/h takes 2.5 hours. Find the speed needed to take 2 hours.
- Write v = k/t.
- Substitute 60 = k ÷ 2.5, so k = 60 × 2.5 = 150.
- The equation is v = 150/t. For t = 2, v = 150 ÷ 2 = 75 km/h.
Check: the distance is 150 km in both cases. 60 × 2.5 = 150 and 75 × 2 = 150.
An inverse-square case
The light intensity I varies inversely as the square of the distance d from a lamp. I = 36 units at d = 2 m. Find I at d = 6 m.
- Write I = k/d².
- Substitute 36 = k ÷ 4, so k = 144.
- For d = 6: I = 144 ÷ 36 = 4 units.
The distance became 3 times as large, so the intensity became 3² = 9 times smaller, and 36 ÷ 9 = 4 agrees.
The mistake that costs marks
The common slip is to write y = kx because the words “varies” sound the same. A student meets “varies inversely”, sees y = 6 when x = 4, and finds k = 1.5.
| Step | Wrong | Right |
|---|---|---|
| Equation | y = kx | y = k/x |
| Find k | 6 = 4k, so k = 1.5 | 6 = k ÷ 4, so k = 24 |
| y when x = 3 | 1.5 × 3 = 4.5 | 24 ÷ 3 = 8 |
The wrong answer goes down when x goes down, which is the opposite of what inverse means. When x decreases from 4 to 3, y must increase from 6. The answer 8 does.
Check yourself
y varies inversely as the square root of x. y = 4 when x = 9. Find y when x = 16.
Answer
Write y = k/√x. Substitute 4 = k ÷ 3, so k = 12.
For x = 16: y = 12 ÷ 4 = 3.
Check: x rose, so y must fall, and 3 is less than 4. Also √x × y = 4 × 3 = 12 = k.
What to study next
Next is working with joint variation, where more than one quantity drives the change. Then practise in the variation practice set and log slips in the mistake log and paper-error review.
For a teacher to go through these with you, see online one-to-one Mathematics tuition.