Proportional Relationships: Constants and Equations | Grade 8 Math | LevelKite

You already know that a unit rate tells how much of one quantity goes with 1 unit of another. Picture an empty tank that fills with water at 2 liters per minute. For the first 5 minutes, let $x$ be the time in minutes since the water started, and let $y$ be the number of liters in the tank.

After 1 minute the tank holds 2 liters. After 3 minutes it holds 6 liters, and after 4.5 minutes it holds 9 liters. Each amount is 2 times its time, so $y = 2x$ . When the time is not 0, the amount divided by the time always equals the unit rate, 2. For example, $\frac{9}{4.5} = 2$ . At time 0, the tank holds $2 \cdot 0 = 0$ liters. Quantities linked by one fixed multiplier in this way are called proportional.

Key concept

The test for a proportional relationship

Two quantities $x$ and $y$ are in a proportional relationship when one fixed number $k$ makes $y = kx$ true for every pair $(x, y)$ . The number $k$ is the constant of proportionality. In a context, $k$ is the unit rate: the amount of $y$ for 1 unit of $x$ .

A table, graph or rule shows a proportional relationship exactly when both checks pass:

Check 1: Every pair with $x \neq 0$ gives the same quotient $\frac{y}{x}$ . That quotient is $k$ .
Check 2: Every pair with $x = 0$ has $y = 0$ , because $y = k \cdot 0 = 0$ . Never divide by 0 here.

This is why the graph of $y = kx$ passes through the origin, the point $(0, 0)$ .

y = kx

x	y
0	0
4	10
6	15
10	25

Worked example

Test a table

Does the table above show a proportional relationship between $x$ and $y$ ? If it does, find the constant of proportionality.

Use check 1 on the pairs whose x-value is not 0: (4, 10), (6, 15) and (10, 25). These x-values are nonzero, so each quotient y divided by x is defined.
$\frac{10}{4} = 2.5, \quad \frac{15}{6} = 2.5, \quad \frac{25}{10} = 2.5$
The three quotients are equal, so check 1 passes. Their shared value is the only number that can be the constant.
$k = 2.5$
Use check 2 on the pair (0, 0). Do not divide, because 0 divided by 0 is undefined. Instead, y = kx says that when x is 0, y must be 2.5 times 0, which is 0. The table gives y = 0, so check 2 passes.
$y = 2.5 \cdot 0 = 0$

Answer

Yes. Both checks pass, so the table shows a proportional relationship, and its constant of proportionality is $k = 2.5$ .

The figure below graphs both tanks for the first 5 minutes. The horizontal axis is $x$ , the time in minutes. The vertical axis is $y$ , the liters in the tank. Each point $(x, y)$ pairs a time with an amount of water.

Line A is the first tank, $y = 2x$ . It passes through the origin $(0, 0)$ . Its marked points $(1, 2)$ and $(2, 4)$ have $x \neq 0$ , and both give the quotient 2: $\frac{2}{1} = 2$ and $\frac{4}{2} = 2$ .
Line B is the second tank, $y = 2x + 3$ . It meets the vertical axis at $(0, 3)$ , not at the origin. Its marked points $(1, 5)$ and $(2, 7)$ have $x \neq 0$ , and their quotients, 5 and 3.5, do not match.

Both graphs are straight lines, but only Line A passes the test. Being a straight line is not enough. Passing through the origin is not enough on its own either. The vertical line through the origin also contains points such as $(0, 1)$ , which fail check 2.

Line A is

y = 2x

and passes through

(0, 0)

. Line B is

y = 2x + 3

and meets the vertical axis at

(0, 3)

.

To write an equation for a proportional relationship, substitute the value of $k$ into $y = kx$ . The equation describes every pair at once: each $y$ is $k$ times its $x$ . In the table example, $k = 2.5$ , so the equation is $y = 2.5x$ . The pair $(6, 15)$ fits, because $2.5 \cdot 6 = 15$ .

The equation and the quotient say the same thing. Dividing both sides of $y = kx$ by the same number $x$ keeps them equal as long as $x \neq 0$ , since division by 0 is undefined. The result is $\frac{y}{x} = k$ . So if you already know that a relationship is proportional, one pair with $x \neq 0$ is enough to find $k$ .

Each point pairs a number of complete laps,

x

, with a distance in kilometers,

y

:

(0, 0)

,

(5, 2)

,

(10, 4)

and

(15, 6)

.

Worked example

Read a graph and write its equation

The figure above shows a runner's distance on a track. The horizontal axis is $x$ , the number of complete laps, and the vertical axis is $y$ , the total distance in kilometers. The labels give the exact points $(0, 0)$ , $(5, 2)$ , $(10, 4)$ and $(15, 6)$ . Is the relationship shown proportional? If so, find $k$ and write an equation.

Check 1: the points whose x-value is not 0 are (5, 2), (10, 4) and (15, 6). Divide each y-value by its nonzero x-value and compare the quotients.
$\frac{2}{5} = 0.4, \quad \frac{4}{10} = 0.4, \quad \frac{6}{15} = 0.4$
Check 2: the point with x = 0 is (0, 0). Its y-value is 0, which is what y = kx requires, since k times 0 is 0. Both checks pass, so the relationship is proportional.
$k = 0.4$
Interpret k as a unit rate: it is the distance for 1 lap. Every lap of the same track has the same length, so each lap is 0.4 kilometer. Since 1 kilometer is 1000 meters, that is 400 meters.
$0.4 \text{ km} = 0.4 \cdot 1000 \text{ m} = 400 \text{ m}$
Write the equation: a proportional relationship has the form y = kx, so substitute k = 0.4.
$y = 0.4x$

Answer

Yes. The relationship is proportional with $k = 0.4$ kilometer per lap, and its equation is $y = 0.4x$ , where $x$ is the number of complete laps and $y$ is the distance in kilometers.

Worked example

Find k from one pair with fractions

In a paint mix, the cups of blue paint, $y$ , are proportional to the cups of yellow paint, $x$ . One batch uses $\frac{3}{4}$ cup of yellow and $\frac{1}{2}$ cup of blue. Find the constant of proportionality, write an equation, and find how much blue paint goes with 6 cups of yellow.

The relationship is proportional, so one pair with a nonzero x-value is enough to find k. Here x is 3/4, which is not 0, so you may divide y by x.
$k = \frac{y}{x} = \frac{1}{2} \div \frac{3}{4}$
Dividing by 3/4 means multiplying by its reciprocal, 4/3, which exists because 3/4 is not 0. Multiply, then simplify.
$k = \frac{1}{2} \cdot \frac{4}{3} = \frac{1 \cdot 4}{2 \cdot 3} = \frac{4}{6} = \frac{2}{3}$
Write the equation: a proportional relationship has the form y = kx, so substitute k = 2/3.
$y = \frac{2}{3}x$
Check the equation with the given pair: when x is 3/4, it should give y = 1/2.
$\frac{2}{3} \cdot \frac{3}{4} = \frac{2 \cdot 3}{3 \cdot 4} = \frac{6}{12} = \frac{1}{2}$
Substitute x = 6 to find the cups of blue paint. Keep k as the exact fraction 2/3. Rounding it to 0.67 would give 4.02, which is only an approximation.
$y = \frac{2}{3} \cdot 6 = \frac{2 \cdot 6}{3} = \frac{12}{3} = 4$

Answer

The constant of proportionality is $\frac{2}{3}$ , so you use $\frac{2}{3}$ cup of blue for each cup of yellow. The equation is $y = \frac{2}{3}x$ , and 6 cups of yellow paint go with exactly 4 cups of blue paint.

Takeaway: A relationship is proportional when one number $k$ makes $y = kx$ true for every pair. To test it, check that every pair with $x \neq 0$ gives the same quotient $\frac{y}{x}$ and that every pair with $x = 0$ has $y = 0$ . Equal steps alone are not enough. That shared quotient is the constant of proportionality, and substituting it into $y = kx$ gives the equation. Next, you will graph proportional relationships and interpret the unit rate $k$ as the slope of the graph, a number that measures its steepness.