Constant of Proportionality

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Constant of Proportionality: meaning, method and solved example (7th Grade) – Maths PDF

In short: The constant of proportionality is the single multiplier k that turns every x into its matching y when two quantities scale together, found as y over x.

7th GradeRatios and Percents7.RP.A.2b

What is Constant of Proportionality?

The constant of proportionality is the one number that never changes in a proportional relationship. If tickets cost the same each, that fixed price per ticket is it, and every total is the number of tickets times that price.

Precisely, in \(y = kx\) the constant is \(k = \frac{y}{x}\). It doubles as the unit rate, the value of y when \(x = 1\), and as the steepness of the graph.

How to spot it

  • A story gives a total for several items and asks for the cost, distance or amount per one.
  • A table of proportional pairs asks for the number you multiply x by to get y.
  • A graph starts at the origin and one point, such as (4, 10), is labeled.
  • An equation y = kx is requested, with k to be found.

The method

  1. Pick any pair \((x, y)\) from the table, the graph or the story.
  2. Divide y by x to get \(k\).
  3. Check with a second pair; it must give the same quotient.
  4. Write the equation \(y = kx\) and say what the constant of proportionality means, with units.
x = 4y = 10(4, 10)k = y / xk = 10 / 4 = 2.5y = 2.5 x
The point (4, 10) gives k = 2.5, the same as the slope of the line.

Constant of Proportionality: solved example

Problem

A school in Kansas City orders 12 tickets to a science museum for $90. Find the price per ticket as the fixed multiplier, then write the cost y of x tickets as an equation.

Solution

Divide the total by the number: \(k = \frac{90}{12} = 7.5\). Each ticket costs $7.50.

The equation is \(y = 7.5x\). Check: 20 tickets cost \(7.5 \cdot 20 = 150\) dollars.

Try it

Question 1

The pairs (3, 21) and (5, 35) are proportional. Find k.

Show the answer

\(k = \frac{21}{3} = 7\), and \(\frac{35}{5} = 7\) confirms it.

Question 2

If \(y = 0.4x\), what is y when \(x = 45\)?

Show the answer

\(0.4 \cdot 45 = 18\).

Constant of Proportionality vs. Slope

For a proportional relationship the two numbers agree, because the line goes through the origin. For other lines the slope still measures steepness, but y divided by x is not constant and no single k exists.

Slope

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