Chapter 3 · Ratios and proportional relationships

Spotting Proportional Relationships in Tables and Graphs

Learn to tell whether a table or a graph shows a proportional relationship by checking for one constant ratio and a straight line through the origin.

18 min read

You already know how to find a unit rate: the amount of one quantity for each 1 unit of another quantity. This lesson asks a bigger question. Does one unit rate fit every pair of values in a relationship?

Suppose a faucet pours 3 liters of water each minute. Let $x$ be the time in minutes and $y$ be the amount of water in liters. After 1 minute there are 3 liters, after 2 minutes there are 6 liters, and after 5 minutes there are 15 liters. Divide the liters by the minutes for each pair:

$3 \div 1 = 3$ , $6 \div 2 = 3$ , $15 \div 5 = 3$

Each pair gives the same result, 3 liters per minute, so one unit rate fits all of them. When that happens, the relationship is called a proportional relationship. In this lesson you will learn to spot one in a table and then on a graph.

Key concept

What proportional means

Let $x$ and $y$ be two quantities that change together. Their relationship is proportional when the ratio $\frac{y}{x}$ (that is, $y \div x$ ) equals one constant $k$ , a number that stays the same, for every pair of values with $x \neq 0$ .

Multiplying both sides of $\frac{y}{x} = k$ by $x$ keeps them equal and gives the rule $y = kx$ . The rule also covers $x = 0$ , where $y = k \cdot 0 = 0$ . You cannot divide by 0, so the ratio is not used for that pair.

On a graph, every pair is then a point on one straight line through $(0, 0)$ and $(1, k)$ . The point $(1, k)$ shows the unit rate: $k$ units of $y$ for each 1 unit of $x$ .

\frac{y}{x} = k \text{ for every pair with } x \neq 0, \text{ so } y = kx

Start with tables. A table lists pairs of values, one pair in each row. To test a table, divide $y$ by $x$ in every row where $x \neq 0$ . (You cannot divide by 0. A row with $x = 0$ fits a proportional relationship only when $y = 0$ , because $y = k \cdot 0 = 0$ .)

Why divide? The result of $y \div x$ is the amount of $y$ for each 1 unit of $x$ , so it is the unit rate for that row. If all the rows you divide give the same result $k$ , one unit rate fits them all: $y = kx$ in each row, and the table shows a proportional relationship. If even one row gives a different result, no single $k$ fits, and the relationship is not proportional.

The table below shows a printer at work. Here $x$ is the time in minutes and $y$ is the number of pages printed. The third column shows the division for each row.

Printer: time and pages printed

Time $x$ (minutes)	Pages printed $y$	Ratio $y \div x$ (pages per minute)
$2$	$12$	$12 \div 2 = 6$
$3$	$18$	$18 \div 3 = 6$
$5$	$30$	$30 \div 5 = 6$
$8$	$48$	$48 \div 8 = 6$

Worked example

Example 1: One constant ratio in a table

The printer table above pairs 2 minutes with 12 pages, 3 minutes with 18 pages, 5 minutes with 30 pages and 8 minutes with 48 pages. Here $x$ is the time in minutes and $y$ is the number of pages printed. Is the relationship between $x$ and $y$ proportional?

State the test. The relationship is proportional when y divided by x equals one constant k in every row where x is not 0. No row has x = 0, so you can divide in all four rows.
$\frac{y}{x} = k \text{ for every row with } x \neq 0$
Divide the pages by the minutes in each row. Each result is the number of pages for 1 minute, so it is the unit rate for that row.
$12 \div 2 = 6, \quad 18 \div 3 = 6, \quad 30 \div 5 = 6, \quad 48 \div 8 = 6$
Compare the four results. Each one is 6, so the same constant, k = 6, fits every row.
$k = 6$
Check the rule y = kx with k = 6. Multiplying each time by 6 gives back the pages in that row, so the rule holds in all four rows.
$6 \cdot 2 = 12, \quad 6 \cdot 3 = 18, \quad 6 \cdot 5 = 30, \quad 6 \cdot 8 = 48$

Answer

Yes, the relationship is proportional. In every row $y \div x = 6$ , so the printer prints 6 pages per minute and $y = 6x$ .

Arcade: games played and total cost

Games played $x$	Total cost $y$ (dollars)	Ratio $y \div x$ (dollars per game)
$1$	$7$	$7 \div 1 = 7$
$2$	$8$	$8 \div 2 = 4$
$3$	$9$	$9 \div 3 = 3$
$4$	$10$	$10 \div 4 = 2.5$

Worked example

Example 2: Ratios that change in a table

An arcade charges 6 dollars to enter plus 1 dollar for each game. The arcade table above shows the total cost: 7 dollars for 1 game, 8 dollars for 2 games, 9 dollars for 3 games and 10 dollars for 4 games. Here $x$ is the number of games played and $y$ is the total cost in dollars. Is the relationship between $x$ and $y$ proportional?

Divide the total cost by the number of games in each row. No row has x = 0, so every division is allowed. Each result is the cost per game for that row.
$7 \div 1 = 7, \quad 8 \div 2 = 4, \quad 9 \div 3 = 3, \quad 10 \div 4 = 2.5$
Compare the results. They are 7, 4, 3 and 2.5, which are not all the same, so no single constant k equals y divided by x in every row. Two rows that disagree are already enough to decide.
$7 \neq 4$
See the failure another way by testing the rule y = kx. If k were 7, as in the first row, then 2 games would cost 14 dollars, but the table shows 8 dollars. The 6-dollar entry fee is spread over more games, so the cost per game keeps changing.
$7 \cdot 2 = 14 \neq 8$

Answer

No, the relationship is not proportional. The ratio $y \div x$ is 7, 4, 3 and 2.5 in the four rows, so it is not one constant.

Now move from tables to graphs. On a graph, each pair of values becomes a point $(x, y)$ . Every value in this lesson is 0 or greater, so no point sits left of the vertical axis or below the horizontal axis.

What does the rule $y = kx$ look like on a graph?

When $x = 0$ , the rule gives $y = k \cdot 0 = 0$ . That is the origin, the point $(0, 0)$ where the two axes meet.
When $x = 1$ , the rule gives $y = k \cdot 1 = k$ . That is the point $(1, k)$ , which shows the unit rate.
Each time $x$ grows by 1, $y$ grows by the same amount $k$ , because $k(x + 1) = kx + k$ . So the points keep one steady slant.

Together, these facts mean that all the points of a proportional relationship lie on one straight line through $(0, 0)$ and $(1, k)$ .

To test a graph, use the exact coordinates of its labeled points instead of judging by eye. Ask two questions:

Does the graph pass through the origin $(0, 0)$ ?
Is $y \div x$ the same constant $k$ at every labeled point with $x \neq 0$ ?

Two yes answers mean every labeled point fits $y = kx$ , so the points lie on one straight line through $(0, 0)$ and $(1, k)$ , and the relationship is proportional. If the points do not lie on one straight line, or if the line misses $(0, 0)$ , the relationship is not proportional.

The graph below shows a lemonade recipe. The horizontal axis shows $x$ , the number of scoops of drink mix. The vertical axis shows $y$ , the number of cups of water. Line A passes through four labeled points.

Line A passes through

(0, 0)

(1, 2)

(3, 6)

and

(5, 10)

, with scoops of mix

x

across and cups of water

y

up.

Worked example

Example 3: Testing a graph

In the graph above, the horizontal axis shows $x$ , the number of scoops of drink mix, and the vertical axis shows $y$ , the number of cups of water. Line A passes through the labeled points $(0, 0)$ , $(1, 2)$ , $(3, 6)$ and $(5, 10)$ . Is the relationship between $x$ and $y$ proportional?

Check the origin. One labeled point on Line A is (0, 0), which says that 0 scoops of mix go with 0 cups of water. That fits the rule y = kx, because k times 0 is 0.
$(0, 0) \text{ is on Line A}$
Find y divided by x at every labeled point where x is not 0. This means dividing the cups of water by the scoops of mix.
$2 \div 1 = 2, \quad 6 \div 3 = 2, \quad 10 \div 5 = 2$
Compare the results. All three are 2, so one constant, k = 2, fits every labeled point. Each point fits the rule y = 2x, so the points lie on one straight line through (0, 0).
$2 \cdot 1 = 2, \quad 2 \cdot 3 = 6, \quad 2 \cdot 5 = 10$
Read the unit rate from the point (1, k). On Line A that point is (1, 2), so 1 scoop of mix goes with 2 cups of water.
$(1, k) = (1, 2)$

Answer

Yes, the relationship is proportional. Line A is a straight line through $(0, 0)$ , and $y \div x = 2$ at every labeled point with $x \neq 0$ , so $y = 2x$ and the unit rate is 2 cups of water per scoop.

A straight line is not enough. A common mistake is to say, "The points make a straight line, so the relationship is proportional." That reasoning would call Line B below proportional. Line B shows the arcade from Example 2, with games played $x$ on the horizontal axis and total cost $y$ in dollars on the vertical axis. It is straight, but it meets the vertical axis at $(0, 6)$ , not at $(0, 0)$ , because 0 games still cost 6 dollars. The ratios disagree as well: $7 \div 1 = 7$ but $8 \div 2 = 4$ . A straight line only shows that $y$ changes by the same amount for each step of 1 in $x$ . Proportional needs more: $y \div x$ must equal one constant $k$ for every pair with $x \neq 0$ . For a straight line, that happens only when it passes through $(0, 0)$ .

Line B passes through

(0, 6)

(1, 7)

(2, 8)

(3, 9)

and

(4, 10)

, so it is straight but misses

(0, 0)

Here is the idea to keep. A relationship between $x$ and $y$ is proportional when $y \div x$ equals one constant $k$ for every pair with $x \neq 0$ , which is the same as $y = kx$ .

In a table, divide $y$ by $x$ in every row with $x \neq 0$ and look for one constant result. A row with $x = 0$ must have $y = 0$ .
On a graph, look for one straight line through $(0, 0)$ and $(1, k)$ , where $(1, k)$ shows the unit rate. A straight line that misses $(0, 0)$ does not show a proportional relationship.

The constant $k$ is called the constant of proportionality. Next, you will learn to find it from a table, from a graph, and from an equation or a description in words.

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Finding the Constant of Proportionality

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