Ratios and Proportions • Topic 5 of 7

Inverse Variation

In inverse variation, two quantities multiply to a constant: y = k/x, equivalently xy = k. As one increases, the other decreases proportionally, so the product stays fixed. To find k, multiply a known pair x·y; then for any new x, divide k by it. Classic examples are "more workers, less time" and "faster speed, less time for a fixed distance." The graph is a curve (a hyperbola), not a line. On the SAT, watch for the words "inversely proportional" or situations where doubling one quantity halves the other — that signals xy = constant.

✅ Solved examples

1. y varies inversely with x, and y = 6 when x = 4. Find k.
k = xy = 4 × 6 = 24.
2. Using xy = 24, find y when x = 8.
y = 24/8 = 3.
3. If 4 workers finish a job in 6 days, how long for 8 workers (same rate)?
Work = 4 × 6 = 24 worker-days; 24 ÷ 8 = 3 days.
4. y is inversely proportional to x; y = 10 when x = 3. Find y when x = 6.
k = 3 × 10 = 30, so y = 30/6 = 5.

✏️ Practice — try these, take hints as needed

1. y varies inversely with x and y = 8 when x = 5. Find k.
k = xy.
5 × 8.
Compute.
40.
2. Using xy = 36, find y when x = 9.
y = k/x.
36 ÷ 9.
Compute.
4.
3. If 6 taps fill a tank in 8 hours, how long for 12 taps?
Total = 6 × 8 = 48 tap-hours.
Divide by 12 taps.
Compute.
4 hours.
4. y ∝ 1/x and y = 12 when x = 2. Find y when x = 8.
k = 2 × 12 = 24.
y = 24/x.
x = 8.
3.
5. A trip takes 5 hours at 60 mph. How long at 75 mph?
Distance = 60 × 5 = 300 miles (constant).
Time = distance ÷ speed.
300 ÷ 75.
4 hours.

📝 Topic test — 8 questions

Auto-graded with full solutions; saved to your dashboard. Use the calculator and formula sheet (top-right) any time.

Loading questions…
← Direct Variation Scale Factors →