Some quantities grow together, and others grow apart. In a direct proportion, y = kx: doubling x doubles y. In an inverse proportion, xy = k: doubling x halves y. Use this calculator to solve both kinds, with the constant, the equation and every step shown.
Direct and Inverse Proportion Calculator
Choose the type, enter a known pair and a new x.
What Is Direct Proportion?
In a direct proportion, the two quantities increase together in the same ratio: y = kx. If you buy twice as many items at the same unit price, you pay twice as much. The ratio y/x is constant, the graph is a straight line through the origin, and the constant k is the unit rate. Examples: distance at a steady speed, total cost at a fixed price, and the amount of paint needed for a wall of a given area.
What Is Inverse Proportion?
In an inverse proportion, one quantity grows as the other shrinks: xy = k. If 4 workers take 20 days to finish a job, 10 workers take 8 days, because 4 × 20 = 10 × 8 = 80 worker-days. The product is constant, the graph is a curve (a hyperbola), and it never touches either axis. Examples: workers and time, speed and travel time for a fixed distance, and pressure and volume of a gas at a fixed temperature.
Direct vs Inverse: How to Tell Them Apart
Ask: when one quantity doubles, what happens to the other? If it doubles too, the proportion is direct. If it halves, the proportion is inverse. In a table, check the ratio y/x for direct proportion (it stays the same) and the product xy for inverse proportion (it stays the same). The classic trap is treating an inverse problem as direct — more workers means less time, not more.
Worked Examples
Example 1 (direct): 4 kg of rice cost $20. What do 10 kg cost? k = 20 ÷ 4 = $5 per kg, so y = 5 × 10 = $50. Example 2 (inverse): 6 pumps empty a tank in 8 hours. How long do 4 pumps take? k = 6 × 8 = 48, so y = 48 ÷ 4 = 12 hours. Example 3 (inverse): Driving at 60 mph takes 2 hours. At 80 mph? k = 60 × 2 = 120, so y = 120 ÷ 80 = 1.5 hours.
Direct and Inverse Proportion FAQs
What is the formula for direct proportion?
y = kx, where k is the constant of proportionality, k = y ÷ x. Solve a direct proportion with the proportion formula a/b = c/d.
What is the formula for inverse proportion?
xy = k, where k is the constant product. If x1 × y1 = x2 × y2, then y2 = (x1 × y1) ÷ x2.
Is speed and time a direct or inverse proportion?
For a fixed distance, speed and time are inversely proportional: drive faster and the trip takes less time. Distance itself is directly proportional to time at a steady speed.
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