Is the constant of proportionality the same as the unit rate?

In a proportional relationship, yes — they are the same number. The unit rate tells you how much y goes with one unit of x, and k is exactly that value, which is why the graph passes through (1, k). The difference is in how they are used: "unit rate" is the language of rates and units (35 pages per minute), while "constant of proportionality" is the language of the equation y = kx.

Can the constant of proportionality be a fraction or a decimal less than 1?

Yes. If y grows more slowly than x, then k is between 0 and 1. For example, if 4 cups of water make 1/2 batch of paste, then k = 1/2/4 = 1/8 batch per cup and y = 1/8x. Nothing about the method changes; only the size of k does. A small k makes the line rise gently, while a large k makes it steep.

Why must a proportional graph go through the origin?

Substitute x = 0 into y = kx: you get y = k · 0 = 0. So the pair (0, 0) always satisfies the equation. In context it makes sense too — buy zero notebooks and you spend zero dollars. If a graph has a starting fee or a head start, it crosses the vertical axis somewhere other than 0 and the ratio y/x will not stay constant.

How do I know which quantity should be x and which should be y?

Look at how the problem is set up. If a graph is given, the horizontal axis label is x and the vertical axis label is y. If a table is given, the problem usually states which column is which, or the phrasing "miles for each gallon" tells you miles depends on gallons. When you get to choose, put the quantity you control on the x-axis. Then always compute k = y/x in that order, and state the units so your meaning is clear.

M7MATH-3.4

Constant of Proportionality & Its Equation

Learn to find the constant of proportionality k from tables, graphs, and words, write y = kx, and explain what any point on the graph means in context.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Constant of Proportionality & Its Equation, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

Start Constant of Proportionality & Its Equation free

What this lesson covers

Every proportional relationship hides a single number that controls everything about it. If 3 pounds of apples cost 6 dollars, that number is 2 — two dollars for every pound. Multiply any number of pounds by 2 and you get the cost. That multiplier is called the constant of proportionality, written

k

.

In this lesson you will pull

k

out of three different places — a table of values, a graph, and a sentence describing a situation — and then turn it into the equation

y = kx

. You will also learn to read a point on the graph out loud in the words of the problem, so that

(4, 10)

stops being just two numbers and becomes "4 hours of work pays 10 dollars." That skill is what makes the rest of the unit, and later work with slope and linear functions, feel like review instead of new material.

What the Constant of Proportionality Actually Is

Two quantities

x

and

y

are proportional when every pair of values has the same quotient:

\frac{y}{x} = k

That fixed quotient

k

is the constant of proportionality. Rearranging gives the equation form you will write over and over:

y = kx

The two forms say the same thing. Use

k = \frac{y}{x}

when you are hunting for the constant, and

y = kx

when you want to predict a value.

The most important habit is deciding which quantity is

y

and which is

x

before you divide, because

k

depends on the order. If a recipe uses 6 cups of flour for 2 loaves, then

\frac{\text{cups}}{\text{loaves}} = \frac{6}{2} = 3 \text{ cups per loaf}

but

\frac{\text{loaves}}{\text{cups}} = \frac{2}{6} = \frac{1}{3} \text{ loaf per cup.}

Both are correct constants of proportionality — for different equations. If

y

stands for cups and

x

for loaves, then

y = 3x

. If

y

stands for loaves and

x

for cups, then

y = \frac{1}{3}x

. This is where students most often go wrong: they divide the larger number by the smaller one out of habit instead of dividing the

y

-value by the

x

-value.

Because

k

is a rate, it always carries units: dollars per pound, miles per hour, cups per loaf. Saying "

k = 3

cups per loaf" shows you understand the number, not just the arithmetic.

Finding k in a Table

In a table, test the quotient

\frac{y}{x}

for every row. If all the quotients match, the relationship is proportional and that common value is

k

Hours worked ( $x$ )	Pay in dollars ( $y$ )	$\frac{y}{x}$
3	40.50	13.50
5	67.50	13.50
8	108	13.50

Every row gives 13.50, so

k = 13.50

dollars per hour and the equation is

y = 13.5x

.

A table can also be missing the easy row. Students sometimes look for the row where

x = 1

and give up when it is not there — but you never need it. Any single row works, because dividing does the same job.

Tables get harder when the values are fractions. Suppose a runner covers

\frac{3}{4}

mile in

\frac{1}{6}

hour. Then

k = \frac{3/4}{1/6} = \frac{3}{4} \cdot \frac{6}{1} = \frac{18}{4} = 4.5 \text{ miles per hour,}

y = 4.5x

where

y

is miles and

x

is hours. The rule does not change; only the division looks messier.

One caution: a table where the quotients are not all equal is not proportional, and it has no constant of proportionality at all. If the rows give 13.50, 13.50, and 14.00, the correct response is "not proportional," not "

k

is about 13.5."

Finding k on a Graph

A proportional relationship graphs as a straight line through the origin. Two points on that line matter especially.

The point

(0, 0)

says that zero of one quantity goes with zero of the other — no hours worked, no pay. Every proportional graph contains it.

The point

(1, k)

is the workhorse. Because

y = kx

, plugging in

x = 1

gives

y = k

. So the

y

-value directly above

x = 1

is the constant of proportionality. If the line passes through

(1, 6)

, then

k = 6

and

y = 6x

.

When the graph has no gridline point at

x = 1

, or the point at

x = 1

falls between grid marks, pick any clearly labeled lattice point and divide. A line through

(4, 10)

gives

k = \frac{10}{4} = 2.5,

y = 2.5x

What you see	What to do	Result
Line through $(1, 9)$	Read the $y$ -value	$k = 9$
Line through $(6, 15)$	Compute $\frac{15}{6}$	$k = 2.5$
Line not through origin	Stop	Not proportional
Curve through origin	Stop	Not proportional

A frequent mistake is reversing the coordinates and computing

\frac{x}{y}

. Always name the axes first: the vertical axis holds

y

, so

k = \frac{y}{x}

means vertical value over horizontal value. Another mistake is reading a point that only looks like it is on the line — choose points that land exactly on grid intersections.

From Words to an Equation, and Back to Meaning

Verbal descriptions usually hand you

k

in a phrase containing "per," "each," "for every," or "at a rate of." "Bananas cost 59 cents per pound" gives

k = 0.59

and

y = 0.59x

with

y

in dollars and

x

in pounds.

When the sentence does not use the word "per," build the ratio yourself. "A printer prints 105 pages in 3 minutes" becomes

k = \frac{105}{3} = 35

pages per minute, so

y = 35x

.

Always write down what

x

and

y

represent, including units. An equation like

y = 35x

is meaningless on its own; "

y

is pages printed and

x

is minutes" makes it usable.

Interpreting a point works in the other direction. For the printer, the point

(4, 140)

means "in 4 minutes the printer prints 140 pages." A complete interpretation names both quantities and both units in the order the axes list them. Two points deserve special sentences:

(0, 0)

: in 0 minutes, 0 pages are printed.

(1, 35)

: in 1 minute, 35 pages are printed — and this is the unit rate, the constant of proportionality itself.

Students most often lose the meaning by saying something like "the point

(4, 140)

means 4 and 140." Push yourself to write a full sentence with nouns. If you can say the sentence out loud and it sounds like something a person would actually say, your interpretation is complete.

Sorting Out the Common Confusions

Three mix-ups account for most wrong answers on this topic.

Reversing the ratio.

k

is always

\frac{y}{x}

, never

\frac{x}{y}

, and never "big number divided by small number." If 4 tickets cost 30 dollars and

y

is cost, then

k = \frac{30}{4} = 7.5

dollars per ticket. If the problem instead defines

y

as tickets and

x

as dollars,

k = \frac{4}{30} = \frac{2}{15}

ticket per dollar. Read the axis labels or the variable definitions before dividing.

Assuming any straight line is proportional. A phone plan costing 20 dollars plus 5 dollars per gigabyte graphs as a straight line, but it misses the origin, so there is no single constant of proportionality. Check for the origin every time.

Adding instead of multiplying to extend a pattern. In the table below, students sometimes see

x

jump by 2 and

y

jump by 7 and then write

y = x + 7

$x$	$y$	$\frac{y}{x}$
2	7	3.5
4	14	3.5
6	21	3.5

The quotient column settles it:

k = 3.5

and

y = 3.5x

. Testing the equation against a row you did not use is a fast self-check. Does

y = 3.5x

give 21 when

x = 6

? Yes, since

3.5 \cdot 6 = 21

. That one substitution catches almost every error before you turn in your work.

Key terms

Constant of proportionality.: The fixed number $k$ that every $x$ -value is multiplied by to get its $y$ -value; equal to $\frac{y}{x}$ for every pair in the relationship.
Proportional relationship.: A relationship between two quantities in which all ratios $\frac{y}{x}$ are equal; its graph is a straight line through the origin.
Equation of a proportional relationship.: The form $y = kx$ , where $k$ is the constant of proportionality and $x$ and $y$ are the two related quantities.
Unit rate.: The amount of $y$ per one unit of $x$ . In a proportional relationship the unit rate equals $k$ , and it appears on the graph at the point $(1, k)$ .
Origin.: The point $(0, 0)$ . Every proportional relationship passes through it, because zero of one quantity pairs with zero of the other.
The point $(1, k)$ .: The point on a proportional graph whose $y$ -value is the constant of proportionality itself, since $y = k \cdot 1 = k$ .
Independent variable ( $x$ ).: The input quantity, graphed on the horizontal axis; the value you choose or that changes on its own.
Dependent variable ( $y$ ).: The output quantity, graphed on the vertical axis; its value depends on $x$ through $y = kx$ .

Worked example

A car's fuel use is recorded in the table below, where

x

is gallons of gas used and

y

is miles driven. Show the relationship is proportional, find the constant of proportionality, write the equation, explain what the point

(1, 31.5)

means, and predict how far the car goes on 12 gallons.

Gallons ( $x$ )	4	6	9
Miles ( $y$ )	126	189	283.5

Step 1 — Test every ratio $\frac{y}{x}$ .

\frac{126}{4} = 31.5 \qquad \frac{189}{6} = 31.5 \qquad \frac{283.5}{9} = 31.5

All three quotients are equal, so the relationship is proportional.

Step 2 — Name the constant with its units.

k = 31.5

miles per gallon. Notice the units come from the division: miles divided by gallons.

Step 3 — Write the equation. Substituting into

y = kx

gives

y = 31.5x,

where

y

is miles driven and

x

is gallons used.

Step 4 — Interpret $(1, 31.5)$ . The first coordinate is gallons and the second is miles, so the point means the car travels 31.5 miles on 1 gallon of gas. This is the unit rate, and it is the same number as

k

— that is always true at

x = 1

.

Step 5 — Predict for 12 gallons. Substitute

x = 12

y = 31.5 \cdot 12 = 378

The car goes 378 miles on 12 gallons.

Step 6 — Check. Twelve gallons is double 6 gallons, and 6 gallons gave 189 miles. Doubling 189 gives 378, which matches. The prediction is consistent with the table.

Practice questions

The graph of a proportional relationship passes through the point

(8, 20)

, where

x

is hours and

y

is dollars earned. What is the equation of the relationship?

$y = 20x$
$y = 2.5x$
$y = 0.4x$
$y = x + 12$

Answer: $y = 2.5x$

The constant of proportionality is

k = \frac{y}{x} = \frac{20}{8} = 2.5

, so the equation is

y = 2.5x

— 2.50 dollars per hour. The choice

y = 0.4x

comes from dividing in the wrong order (

\frac{8}{20}

), which would answer a different question: hours per dollar. The choice

y = 20x

misreads the point as if

x

were 1. And

y = x + 12

is not proportional at all, since its graph misses the origin. Check the winner:

2.5 \cdot 8 = 20

, matching the given point.

A table shows that 5 notebooks cost 8.75 dollars and 8 notebooks cost 14 dollars. Find the constant of proportionality with its units, write the equation, and explain what the point

(3, 5.25)

would mean on the graph.

Answer: $k = 1.75$ dollars per notebook; $y = 1.75x$ ; the point $(3, 5.25)$ means 3 notebooks cost 5.25 dollars.

Divide cost by number of notebooks for each pair:

\frac{8.75}{5} = 1.75

and

\frac{14}{8} = 1.75

. The quotients agree, so the relationship is proportional with

k = 1.75

dollars per notebook, giving

y = 1.75x

where

y

is cost in dollars and

x

is notebooks. To interpret

(3, 5.25)

, match each coordinate to its axis: the first is notebooks, the second is cost. Verify with the equation:

1.75 \cdot 3 = 5.25

, so the point does lie on the line. A full interpretation names both quantities and both units, not just the numbers.

Kira says that because the line through

(0, 4)

and

(2, 10)

is straight, it must be proportional with

k = 3

. Is she right? Explain.

Answer: No. The line does not pass through the origin, so it is not a proportional relationship and it has no constant of proportionality.

Being straight is not enough — a proportional graph must be a straight line through

(0, 0)

. Testing the ratio confirms the problem: at

(2, 10)

the quotient is

\frac{10}{2} = 5

, but at a point like

(4, 16)

on the same line the quotient is

\frac{16}{4} = 4

. Since

\frac{y}{x}

changes, no single

k

works. The 3 Kira found is the rate of change (the

y

-value climbs 6 while

x

climbs 2), which is a real feature of the line but is not a constant of proportionality here because of the starting value of 4.

FAQ

Is the constant of proportionality the same as the unit rate?: In a proportional relationship, yes — they are the same number. The unit rate tells you how much $y$ goes with one unit of $x$ , and $k$ is exactly that value, which is why the graph passes through $(1, k)$ . The difference is in how they are used: "unit rate" is the language of rates and units (35 pages per minute), while "constant of proportionality" is the language of the equation $y = kx$ .
Can the constant of proportionality be a fraction or a decimal less than 1?: Yes. If $y$ grows more slowly than $x$ , then $k$ is between 0 and 1. For example, if 4 cups of water make $\frac{1}{2}$ batch of paste, then $k = \frac{1/2}{4} = \frac{1}{8}$ batch per cup and $y = \frac{1}{8}x$ . Nothing about the method changes; only the size of $k$ does. A small $k$ makes the line rise gently, while a large $k$ makes it steep.
Why must a proportional graph go through the origin?: Substitute $x = 0$ into $y = kx$ : you get $y = k \cdot 0 = 0$ . So the pair $(0, 0)$ always satisfies the equation. In context it makes sense too — buy zero notebooks and you spend zero dollars. If a graph has a starting fee or a head start, it crosses the vertical axis somewhere other than 0 and the ratio $\frac{y}{x}$ will not stay constant.
How do I know which quantity should be $x$ and which should be $y$ ?: Look at how the problem is set up. If a graph is given, the horizontal axis label is $x$ and the vertical axis label is $y$ . If a table is given, the problem usually states which column is which, or the phrasing "miles for each gallon" tells you miles depends on gallons. When you get to choose, put the quantity you control on the $x$ -axis. Then always compute $k = \frac{y}{x}$ in that order, and state the units so your meaning is clear.

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