Proportion.
Turn a proportional relationship into an equation and discover the constant that connects it.
Model direct and inverse proportion · use powers and roots · solve for unknown quantities
Turn ∝ into an equation
Imagine buying identical notebooks at a fixed unit price. Doubling the number bought doubles the total cost. This is direct proportion: the two quantities change in the same multiplicative ratio.
The proportionality sign describes a relationship, but an equation needs a constant multiplier k. In y = kx, k is the amount of y per unit x. In inverse proportion, increasing one quantity instead reduces the other so that a specified product stays constant.
“Proportional to” means a constant multiplier connects the quantities. Replace ∝ with = and a constant k, then use a known pair to find k.
| Relationship | Equation | Effect of doubling x |
|---|---|---|
| y ∝ x | y = kx | y doubles |
| y ∝ x² | y = kx² | y multiplies by 4 |
| y ∝ √x | y = k√x | y multiplies by √2 |
| y ∝ x³ | y = kx³ | y multiplies by 8 |
| y ∝ ∛x | y = k∛x | y multiplies by ∛2 |
| y ∝ 1x² | y = kx² | y divides by 4 |
Pause and explainIf y is directly proportional to x, what happens to y when x triples?
It triples too, because the constant multiplier k stays unchanged.
Find the constant, then the unknown
The proportionality description tells you the shape of the relationship; one known pair of values tells you the constant needed for this particular situation. Keep those two jobs separate.
Write the equation with k first, substitute the known pair to find k, then use the completed equation for the new input. Using a ratio shortcut before identifying whether the relationship is direct, inverse or a power can apply the wrong scaling.
y is directly proportional to x². When x = 3, y = 45. Find y when x = 5.
When x = 5, y = 5 × 25 = 125.
y is inversely proportional to x. When x = 4, y = 15. Find x when y = 10.
10 = 60x, so x = 6.
Pause and explainWhy find k before substituting the new value?
The known pair determines the actual relationship. Once k is fixed, the same equation connects every permitted pair.
Root and inverse-power relationships
“Proportional to the square” means scaling follows x², not x. If x doubles, x² becomes four times as large. A square-root relationship changes more slowly, because the root reverses squaring.
An inverse power combines these ideas: if y is inversely proportional to x², doubling x divides y by four. Translate the full phrase into an equation before calculating, and check how a simple doubling should affect the result.
y ∝ √x and y = 12 when x = 9. Find x when y = 20.
12 = k√9 = 3k, so k = 4. Then 20 = 4√x, so √x = 5 and x = 25.
y ∝ 1∛x and y = 6 when x = 8. Find y when x = 27.
6 = k ÷ 2, so k = 12. At x = 27, y = 12 ÷ 3 = 4.
Pause and explainIf y is proportional to √x, what happens when x is multiplied by 9?
The square root is multiplied by 3, so y is multiplied by 3, not 9.
Your turn. Write the model first.
Underline every word in the proportionality statement, including “inverse”, “square”, “cube” or “root”. Use all of them when writing the equation.
After finding k, predict whether the new answer should increase or decrease. Substitute the original known pair back into your completed equation as a check before using the new input.
y ∝ x³. When x = 2, y = 40. Find y when x = 3.
Reveal worked answer
y is inversely proportional to x². When x = 3, y = 8. Find y when x = 6.
Reveal worked answer
y ∝ ∛x. When x = 8, y = 10. Find x when y = 20.