Constant of Proportionality (k)
--
Equation
--
Calculates the constant 'k' based on the relationship between x and y.

This tool helps you find the constant of proportionality k. It works for both direct proportion and inverse proportion.

Formulas

Direct proportion

If y ∝ x, then y = kx and k = y ÷ x (with x ≠ 0).

Inverse proportion

If y ∝ 1/x, then y = k/x and k = x × y (with x ≠ 0).

How to use

  1. Choose type: Direct (y = kx) or Inverse (y = k/x).
  2. Enter a matching pair of values (x, y).
  3. Compute k with the correct formula.
  4. Use the value of k to make predictions for other x or y.

Quick examples

Example 1 — Direct proportion

Given x = 8 and y = 56. k = y ÷ x = 56 ÷ 8 = 7. Model: y = 7x. For x = 12, y = 84.

Example 2 — Inverse proportion

Given x = 5 and y = 9.6. k = x × y = 5 × 9.6 = 48. Model: y = 48/x. For x = 12, y = 4.

Example 3 — Using two points (direct)

Points: (x1, y1) = (3, 12) and (x2, y2) = (9, 36). Check: y/x is constant? 12/3 = 4, 36/9 = 4. So k = 4 and y = 4x.

Unit notes

  • In direct proportion, k has units (units of y)/(units of x).
  • In inverse proportion, k has units (units of x × units of y).
  • Keep units consistent to avoid errors.

Common mistakes

  • Using the direct formula when the relation is inverse, or the other way around.
  • Mixing units (cm with m, minutes with hours).
  • Dividing by zero or using an x value of zero.
  • Assuming proportionality without checking if y/x (direct) or x·y (inverse) is constant.

FAQ

What is the constant of proportionality?

It is the fixed number k that links two variables in a proportional relation.

How do I know if it is direct or inverse?

For direct, the ratio y/x stays the same. For inverse, the product x·y stays the same.

Can I find k from one data pair?

Yes, if you know the relation type. Use k = y/x (direct) or k = x·y (inverse).

What if I have two points but do not know the type?

Test both: if y1/x1 = y2/x2, it is direct. If x1·y1 = x2·y2, it is inverse.

What happens if x = 0?

The formulas break. Proportional models need nonzero x.

Can k be negative?

Yes. A negative k means y and x move in opposite directions for the chosen model.

Do I need a line through the origin for direct proportion?

Yes. Direct proportion graphs are straight lines through (0, 0).

How do I predict new values?

Direct: y = kx. Inverse: y = k/x. Plug in the new x.

Can I use decimals?

Yes. The formulas work with integers, decimals, and fractions.

Is correlation the same as proportionality?

No. Proportionality is a strict math relation. Correlation is a statistical link.

How do I get k from a graph?

For direct, find the slope of the line through the origin. That slope is k.

What if my data is noisy?

Use best-fit methods (like linear regression through the origin) to estimate k.