Constant of Proportionality Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Constant of Proportionality.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The constant ratio k between two proportional quantities: if y = kx, then k is the constant of proportionality.

If y is always 3 times x, the constant of proportionality is 3.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: In y = kx, k is the multiplier that connects x to y.

Common stuck point: Finding k from a table: compute k = \frac{y}{x} for any row—if all rows give the same k, it's proportional.

Sense of Study hint: Pick any row from the table and divide y by x -- if the ratio is the same for every row, that ratio is k.

Worked Examples

Example 1

easy
A car travels 60 miles per hour. Write the equation relating distance \(d\) and time \(t\). What is the constant of proportionality?

Solution

  1. 1
    The relationship is \(d = k \cdot t\) where \(k\) is the constant of proportionality.
  2. 2
    Here, speed = 60 mph, so \(k = 60\).
  3. 3
    Equation: \(d = 60t\).
  4. 4
    In 3 hours: \(d = 60 \times 3 = 180\) miles.

Answer

\(d = 60t\); \(k = 60\)
In \(y = kx\), \(k\) is the constant of proportionality — the unit rate. Here \(k = 60\) miles per hour.

Example 2

medium
The table shows \(x\) and \(y\): (2, 10), (4, 20), (6, 30). Is this proportional? Find \(k\).

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Apples cost \$0.75 each. Write the equation for cost \(C\) given quantity \(q\). Find the cost of 8 apples.

Example 2

medium
If \(y = kx\) and \(y = 35\) when \(x = 7\), find \(k\) and predict \(y\) when \(x = 12\).
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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