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\(k\)
\(y\) = \(kx\)

The Constant of
Proportionality

Learn to identify the constant of proportionality (unit rate) in tables, graphs, equations, and verbal descriptions of proportional relationships.

7th Grade Math

What Is a Proportional Relationship?

A proportional relationship exists between two quantities when they increase or decrease at the same rate. The ratio between them always stays constant.

Proportional
Ratio \(\frac{y}{x}\) is the same for every pair
Not Proportional
Ratio \(\frac{y}{x}\) changes from pair to pair

Defining \( k \): The Constant

The constant of proportionality is the unchanging ratio between two proportional quantities. We call it \( k \).

When you rearrange, you get the equation of every proportional relationship:

Why Does \( k \) Matter?

The constant of proportionality tells you the rate of change in real-world scenarios: speed, price per item, or conversion factors.

Speed
\( k = 60 \) mph means you travel 60 miles for every 1 hour
Price
\( k = 3.50 \) per lb means each pound costs $3.50
Conversion
\( k = 2.54 \) cm/in converts inches to centimeters

The Core Rule

x y 1 2 3 1 2 3 4
\((1, 1.5)\)
\((2, 3)\)
\((3, 4.5)\)
\(k = 1.5\)
\((0, 0)\)

Finding \( k \) from a Table

\( x \) \( y \) \( \frac{y}{x} \)
263
5153
8243
10303

Finding \( k \) from a Graph

x y 1 2 3 4 2 4 6 8
\((1, 2)\)
\( \text{run} = 1 \)
\( \text{rise} = 2 \)
\(k = 2\)

Finding \( k \) from an Equation

Any equation in the form \( y = kx \) already tells you the constant of proportionality. It is the number multiplying \( x \).

\( y = 7x \)
\( k = \) 7
\( y = 0.5x \)
\( k = \) 0.5
\( y = \frac{3}{4}x \)
\( k = \) \(\frac{3}{4}\)

Proportional vs. Linear

Proportional Linear (not proportional)
Equation\( y = kx \)\( y = mx + b \), \( b \neq 0 \)
Passes through origin?Yes, alwaysNo
Ratio \(\frac{y}{x}\)Constant for all pointsChanges from point to point
Example\( y = 4x \)\( y = 4x + 3 \)

Finding \( k \) from Words

Verbal descriptions often hide the constant of proportionality in phrases like "per", "each", "for every", or "at a rate of".

"$3 per pound"
\( k = 3 \) dollars per pound
"15 miles each hour"
\( k = 15 \) miles per hour

How to Find \( k \): Step by Step

1
Identify the two quantities. Decide which is \( x \) (input) and which is \( y \) (output).
2
Compute the ratio. Divide \( y \div x \) for any given pair of values.
3
Verify consistency. Check that the ratio \( \frac{y}{x} \) is the same for every data pair.
4
State the constant. If the ratio is constant, that value is \( k \). Write the equation \( y = kx \).

Example 1: Table

A bakery sells cupcakes. The table shows the total cost for different numbers of cupcakes. Find the constant of proportionality.

Cupcakes (\( x \)) Cost (\( y \))
3$7.50
6$15.00
10$25.00

Example 2: Graph

A runner's distance over time is graphed below. Find \( k \).

Hours Miles 1 2 3 4 6 12 18 24 30
\((1, 6)\)
\(k = 6\)

Example 3: Verbal Description

1
Identify quantities: \( x \) = hours, \( y \) = widgets.
2
Spot the keyword: "24 widgets per hour" gives the unit rate directly.
3
State the constant: \( k = 24 \). Equation: \( y = 24x \).

Common Mistakes to Avoid

✕ Dividing \( x \div y \)
The constant is \( k = \frac{y}{x} \), not \( \frac{x}{y} \). Flipping the fraction gives the reciprocal, which is a different rate.
✕ Ignoring the origin
A graph that does not pass through \((0, 0)\) is not proportional, even if it looks like a straight line.
✕ Checking only one pair
One matching ratio is not enough. You must verify that \( \frac{y}{x} \) is the same for all pairs in a table.

Finding \( k \): Summary

Representation How to Find \( k \) Quick Tip
TableDivide \( \frac{y}{x} \) for each rowAll ratios must be equal
GraphRead \( y \) when \( x = 1 \)Line must pass through \((0,0)\)
EquationRead the coefficient of \( x \)Must be \( y = kx \) form (no \( + b \))
WordsFind the number before "per" or "each"The word after "per" is the \( x \)-unit
DiagramExtract a ratio from labeled valuesSet up \( \frac{y\text{-quantity}}{x\text{-quantity}} \)

Question 1 of 5

The equation \( y = 4.5x \) describes a proportional relationship. What is the constant of proportionality?

Question 2 of 5

A table shows: \((2, 10)\), \((4, 20)\), \((7, 35)\). What is \( k \)?

Question 3 of 5

A graph shows a straight line through the origin. The point \((1, 8)\) lies on the line. What is the constant of proportionality?

Question 4 of 5

"A car travels at a constant speed of 60 miles per hour." Which equation represents this proportional relationship?

Question 5 of 5

Which table does NOT show a proportional relationship?