Direct proportion
Two quantities are directly proportional when one is a fixed number times the other: y = k·x. The number k is the constant of proportionality. In simple words: if x becomes 2, 3 or 10 times bigger, y also becomes 2, 3 or 10 times bigger.
Test: divide y by x for every pair. If you always get the same number, the relation is direct. Its graph is a straight line through the origin (because x = 0 gives y = 0).
Example: if 1 notebook costs ₹12, then y = 12x. 5 notebooks cost ₹60.
Inverse proportion
Two quantities are inversely proportional when their product stays fixed: x·y = k, so y = k/x. If x becomes 2 times bigger, y becomes half. The graph is a smooth curve (a hyperbola) that never touches either axis, because x and y can never be 0.
Careful: "y goes down when x goes up" is not enough. y = 10 − x also goes down, but x·y is not constant. Always check the product.
Example: 6 workers finish a job in 10 days. The work is 6 × 10 = 60 worker-days. 15 workers need 60 ÷ 15 = 4 days.
Linear versus directly proportional
Every direct proportion is linear, but not every linear relation is a direct proportion. A linear relation is y = a·x + b. It is directly proportional only if b = 0, which means the line goes through the origin.
A taxi that charges ₹20 plus ₹12 per km: 10 km costs ₹140 and 20 km costs ₹260. Doubling the distance did not double the fare, because of the fixed ₹20. The graph starts at 20 on the y-axis, not at 0.
Power relations y = c·xⁿ
A power relation has the form y = c·xⁿ, where c is a constant and n is the exponent (it can be a whole number, a fraction or negative).
- n = 1 gives direct proportion y = c·x.
- n = −1 gives inverse proportion y = c/x.
- n = 2: area of a square or circle (y = side², y = πr²).
- n = 3: volume of a cube (y = side³).
- n = 1/2: y = c√x, for example the time a dropped stone needs depends on √height.
Finding the exponent n and the constant c
Suppose you know two points (x₁, y₁) and (x₂, y₂) of y = c·xⁿ.
- Divide the two equations: y₂/y₁ = (x₂/x₁)ⁿ. The constant c disappears.
- Find n: n = log(y₂/y₁) ÷ log(x₂/x₁). If the numbers are friendly, you can see it: x ratio 2 and y ratio 4 means n = 2.
- Find c: c = y₁ ÷ x₁ⁿ.
- Check with a third point.
Log-log trick: taking logs gives log y = log c + n·log x. So if you plot log y against log x, you get a straight line with slope n. That is how scientists find exponents from data.
The effect of scaling
If x is multiplied by a factor f, then in y = c·xⁿ the value y is multiplied by fⁿ. The constant c does not matter.
- Direct (n = 1): factor f.
- Inverse (n = −1): factor 1/f.
- Area (n = 2): factor f². Double the side, 4 times the area.
- Volume (n = 3): factor f³. Double the side, 8 times the volume.
This is why a 1:10 scale model has 1/100 of the surface area and 1/1000 of the volume of the real thing.
Try it
In the 3D: in the last step set c = 1 and n = 2. Before you look, guess: what is y at x = 3? Then check the dot. Now set n = −1 and guess again.
At home: take 3 squares of paper with sides 5 cm, 10 cm and 20 cm. Count how many small 5 cm squares cover each one: 1, 4 and 16. Side doubled, area four times: n = 2.
Key formulas and definitions
- Direct proportion: y = k·x, y/x = k (constant)
- Inverse proportion: y = k/x, x·y = k (constant)
- Linear: y = a·x + b; proportional only if b = 0
- Power relation: y = c·xⁿ
- Scaling: x → f·x gives y → fⁿ·y
- From two points: n = log(y₂/y₁) / log(x₂/x₁), c = y₁ / x₁ⁿ
Worked examples
1. 5 notebooks cost ₹60. Find the cost of 8 notebooks.
Cost ÷ number = 60 ÷ 5 = 12 is constant, so y = 12x. For 8 notebooks: 12 × 8 = ₹96.
2. 6 workers finish a job in 10 days. How long will 15 workers (same speed) take?
More workers, fewer days: inverse. x·y = 6 × 10 = 60. For 15 workers: 60 ÷ 15 = 4 days.
3. A taxi charges ₹20 plus ₹12 per km. Is the fare directly proportional to the distance?
Fare = 12x + 20. Here b = 20, not 0. Check: 10 km → ₹140, 20 km → ₹260, which is not double of 140. So it is linear but not directly proportional.
4. The area of a circle is A = πr². By what factor does the area grow if the radius becomes 3 times bigger?
n = 2 and f = 3, so the area is multiplied by 3² = 9.
5. A power relation y = c·xⁿ passes through (2, 12) and (4, 48). Find n and c.
y₂/y₁ = 48/12 = 4 and x₂/x₁ = 2. So 2ⁿ = 4, n = 2. c = 12 ÷ 2² = 3. The relation is y = 3x².
6. A relation y = c·xⁿ passes through (2, 5) and (6, 45). Find y at x = 10.
y ratio = 45/5 = 9, x ratio = 3, so 3ⁿ = 9 and n = 2. c = 5 ÷ 2² = 1.25. At x = 10: y = 1.25 × 100 = 125.
7. A model ship is built at 1:50 in length. The real ship needs 2,500 m² of paint. How much paint does the model need?
Area scales with the square of the length factor: (1/50)² = 1/2500. Paint for the model = 2500 ÷ 2500 = 1 m².
Common mistakes
- Thinking every straight line is a direct proportion. Check that the line goes through the origin (b = 0).
- Using "y decreases when x increases" as the test for inverse proportion. The product x·y must be constant.
- Multiplying by f instead of fⁿ when scaling area (n = 2) or volume (n = 3).
- Subtracting the y values instead of dividing them when finding n. Use the ratios: n = log(y ratio) ÷ log(x ratio).