Pre-Algebra
Proportional relationships
Constant of proportionality
20 practice questions
0 video lessons
Theory + worked examples
Constant of Proportionality
California Pre-Algebra • Standard 7.RP.2b • Proportional Relationships
Constant of Proportionality is a topic in Proportional Relationships in the California Common Core State Standards. It is aligned to Standard 7.RP.2b, which requires students to identify the constant of proportionality in tables, graphs, and equations.
The constant of proportionality \(k\) is the constant ratio \(\dfrac{y}{x}\) — the unit rate.
Theory
The constant of proportionality \(k\) is the constant ratio \(\dfrac{y}{x}\) in a proportional relationship.
\(k\) is the unit rate — the value of \(y\) when \(x=1\).
\(k\) is the value at \(x=1\).
What \(k\) means.
Definition:
\[k=\dfrac{y}{x},\qquad y=kx\]
Read \(k\) off the point \((1,k)\).
How to find k
- Pick a point \((x,y)\) on the line.
- Divide \(y\) by \(x\).
- That quotient is \(k\).
- Write \(y=kx\).
Example 1 — From a point
A line through the origin passes \((4,12)\). Find \(k\).
Solution
Divide.
| \(k\) | \(=\) | \(\dfrac{12}{4}=3\) |
Example 2 — From a table
A table shows \(3\) lb costs \(\$7.50\). Find \(k\) (price per lb).
Solution
Unit rate.
| \(k\) | \(=\) | \(\dfrac{7.50}{3}=\$2.50\) |
Example 3 — Write the equation
If \(k=6\), write the equation.
Solution
Use \(y=kx\).
| \(y\) | \(=\) | \(6x\) |
Example 4 — Interpret
In \(y=2x\), what does \(k=2\) mean for \((1,2)\)?
Solution
When \(x=1\), \(y=2\) — the unit rate.
Common pitfalls
\(k=\dfrac{y}{x}\), not \(\dfrac{x}{y}\).
\(k\) is the unit rate, the value at \(x=1\).
Only proportional relationships have a single \(k\).
Frequently asked questions
What is the constant of proportionality?
The constant ratio \(y/x\).
How do I find it?
Divide \(y\) by \(x\) for any point.
What does it represent?
The unit rate.
Line through \((4,12)\): find \(k\).
\(3\).
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