Compare Proportional Relationships and Solve Problems | Grade 8 Math | LevelKite

Two printers or two farm stands can each be described in a different way: one with a table, another with a graph, an equation or a sentence. To decide which one is faster or cheaper, you need a fair way to compare them.

You already know that a proportional relationship between two quantities $x$ and $y$ can be written as $y = kx$ , where $k$ is a fixed number called the constant of proportionality. When $x = 0$ , $y = 0$ . For every pair of values with $x \neq 0$ , the ratio $\frac{y}{x}$ equals $k$ . The graph lies on a non-vertical straight line through the origin $(0, 0)$ , and the slope of that line is $k$ . The number $k$ is also the unit rate: the amount of $y$ for each $1$ unit of $x$ .

In this lesson you will compare proportional relationships that are given in different forms. Then you will use $y = kx$ to solve word problems.

Key concept

One unit rate, many forms

A proportional relationship $y = kx$ has one unit rate $k$ , and every form of the relationship holds that same $k$ .

Equation: in $y = kx$ , $k$ is the number that multiplies $x$ .
Table: divide $y$ by $x$ in any row with $x \neq 0$ .
Graph: divide $y$ by $x$ at any point on the line with $x \neq 0$ . This ratio is the slope of the line.
Words: divide the amount of $y$ by the matching amount of $x$ , which must not be zero.

To compare two proportional relationships, find $k$ for each one in the same units, such as pages per minute for both. Then compare the two values of $k$ . Each unit rate describes the same amount of $x$ , exactly $1$ unit, so the comparison is fair.

k = \frac{y}{x} \quad \text{for } x \neq 0

Check first. Dividing $y$ by $x$ gives the unit rate $k$ only when the relationship is proportional and $x \neq 0$ . Division by zero is undefined, so you cannot use the pair $(0, 0)$ to find $k$ .

Not every straight line is proportional. The line $y = 2x + 3$ crosses the $y$ -axis at $3$ , not at the origin, and its ratio $\frac{y}{x}$ changes: it is $\frac{5}{1} = 5$ at $x = 1$ but $\frac{9}{3} = 3$ at $x = 3$ . This relationship is linear, meaning its graph is a straight line, but it is not proportional. A vertical line is not proportional either. Its slope is undefined, because the change in $x$ is $0$ and you cannot divide by zero. It is also not a function: a function gives exactly one $y$ -value for each $x$ -value, but a vertical line gives many.

Printer A: pages printed after different times

Time in minutes (x)	Pages printed (y)
$2$	$36$
$5$	$90$
$8$	$144$

Worked example

Table and equation: which printer is faster?

Two printers each print at a constant rate, starting from $0$ pages, so each relationship is proportional. Let $x$ be the time in minutes, with $x \ge 0$ , and let $y$ be the number of pages printed. The table above describes Printer A. The equation $y = 20x$ describes Printer B. Call their unit rates $k_A$ and $k_B$ . Which printer is faster, and by how many pages per minute?

Decide what to compare. For each printer, the unit rate is the number of pages for each 1 minute, so both unit rates are in pages per minute. Matching units make the comparison fair.
Find Printer A's unit rate from the table. Divide y by x in each row. Every x in the table is nonzero, so each division is defined. All three rows give the same ratio, as a proportional relationship requires.
$k_A = \frac{36}{2} = \frac{90}{5} = \frac{144}{8} = 18$
Read Printer B's unit rate from its equation. In y = kx, the number that multiplies x is k. In y = 20x, that number is 20.
$k_B = 20$
Compare the unit rates. Both are exact values in pages per minute, and 20 is greater than 18, so Printer B is faster. Subtract to find by how much.
$k_B - k_A = 20 - 18 = 2$

Answer

Printer B is faster. It prints $20$ pages per minute, which is $2$ pages per minute more than the $18$ pages per minute of Printer A.

Now compare a graph with a sentence. Two farm stands sell apples by weight, and at each stand the cost is proportional to the weight. Let $x$ be the weight in pounds, with $x \ge 0$ , and let $y$ be the cost in dollars.

In the figure below, the horizontal axis shows $x$ and the vertical axis shows $y$ . Line A is the graph for Stand A. It passes through the origin and the marked point $(4, 7)$ , so $4$ pounds cost $7$ dollars at Stand A. Stand B gives its price in words: $8$ pounds cost $10$ dollars. Line B shows that same relationship on the same axes. It passes through the origin and the marked point $(8, 10)$ .

Use the exact marked coordinates as your evidence, not how steep a line looks.

Line A passes through

(0, 0)

and

(4, 7)

. Line B passes through

(0, 0)

and

(8, 10)

.

Worked example

Graph and sentence: which stand is cheaper?

Use the figure above. At two farm stands, the cost $y$ in dollars is proportional to the weight $x$ in pounds of apples, with $x \ge 0$ . Line A is Stand A's graph and passes through $(0, 0)$ and $(4, 7)$ . Stand B charges $10$ dollars for $8$ pounds. Call their unit rates $k_A$ and $k_B$ . Which stand has the lower price per pound, and by how much?

Find Stand A's unit rate from the graph. Line A is a non-vertical straight line through the origin, so it shows a proportional relationship. Its marked point (4, 7) has an x-coordinate that is not zero, so divide the y-coordinate by the x-coordinate. This ratio is also the slope of Line A, in dollars per pound.
$k_A = \frac{7}{4} = 1.75$
Find Stand B's unit rate from the sentence. Divide the cost, 10 dollars, by the matching weight, 8 pounds. The divisor 8 is not zero. Simplify the fraction by dividing 10 and 8 by 2.
$k_B = \frac{10}{8} = \frac{5}{4} = 1.25$
Compare the unit rates. Both are exact values in dollars per pound, and 1.25 is less than 1.75, so Stand B charges less for each pound. Subtract to find the difference.
$k_A - k_B = 1.75 - 1.25 = 0.5$
Check against the figure. Line B is the graph of y = 1.25x, and it passes through (8, 10) because 1.25 times 8 is 10. Both unit rates are positive and both lines are on the same axes, so the line with the smaller unit rate, Line B, is less steep. Notice that (8, 10) has the larger y-coordinate, yet Stand B has the smaller unit rate.
$1.25 \cdot 8 = 10$

Answer

Stand B has the lower price. It charges $1.25$ dollars per pound, which is $0.50$ dollars per pound less than the $1.75$ dollars per pound at Stand A.

A word problem about a proportional relationship usually gives you one pair of values and asks you for another. This plan keeps you on track:

Define $x$ and $y$ with their units, and state which values $x$ can take.
Check that the situation is proportional: $y$ is $0$ when $x$ is $0$ , and $y$ changes at a constant rate. A cost of $10$ dollars to join plus $5$ dollars per visit fails this check, because $0$ visits still cost $10$ dollars.
Find $k$ by dividing a known $y$ by its matching nonzero $x$ .
Write $y = kx$ and substitute the value you know.
Answer in a sentence with units.

Worked example

Word problem: downloading a game update

A game update downloads at a constant rate, starting from $0$ gigabytes. After $4$ minutes, $6$ gigabytes have downloaded. How many gigabytes have downloaded after $10$ minutes? The full update is $18$ gigabytes. How many minutes does the full download take?

Define the variables. Let x be the time in minutes since the download started, with x at least 0, and let y be the number of gigabytes downloaded. The download starts at 0 gigabytes and its rate is constant, so y is proportional to x, with unit rate k.
$y = kx$
Find the unit rate from the known pair: 6 gigabytes in 4 minutes. The divisor 4 is not zero. Simplify the fraction by dividing 6 and 4 by 2. The decimal 1.5 is exact, and its unit is gigabytes per minute.
$k = \frac{6}{4} = \frac{3}{2} = 1.5$
Write the equation for this download by replacing k with 3/2.
$y = \frac{3}{2}x$
Answer the first question. Substitute 10 for x, then multiply.
$y = \frac{3}{2} \cdot 10 = \frac{30}{2} = 15$
Begin the second question. Substitute 18 for y. Now x is the unknown.
$18 = \frac{3}{2}x$
Multiply both sides by 2/3. This is the reciprocal of 3/2: the number that multiplies with 3/2 to make 1. It exists because 3/2 is not zero. Multiplying both sides by the same number keeps the equation true, and the right side becomes 1 times x, which is x.
$x = 18 \cdot \frac{2}{3} = \frac{36}{3} = 12$

Answer

After $10$ minutes, $15$ gigabytes have downloaded. The full $18$ -gigabyte update takes $12$ minutes to download.

Line C is the graph of

y = \frac{3}{2}x

. It passes through

(0, 0)

,

(4, 6)

,

(10, 15)

and

(12, 18)

.

The figure above shows the download as a graph. The horizontal axis shows the time $x$ in minutes, and the vertical axis shows the amount downloaded $y$ in gigabytes. Line C is the graph of $y = \frac{3}{2}x$ . The point $(4, 6)$ is the pair given in the problem. The points $(10, 15)$ and $(12, 18)$ are the two answers. The download is finished at $x = 12$ , so only the part of Line C from $(0, 0)$ to $(12, 18)$ describes it.

You can check the answers with the ratio $\frac{y}{x}$ . The points $(4, 6)$ , $(10, 15)$ and $(12, 18)$ all have $x \neq 0$ , and they give the same exact value: $\frac{6}{4} = \frac{15}{10} = \frac{18}{12} = 1.5$ . A constant ratio shows that all three pairs fit the same proportional relationship, at $1.5$ gigabytes per minute.

Takeaway. A proportional relationship $y = kx$ keeps the same unit rate $k$ in every form: a table, a graph, an equation or a sentence. To compare two proportional relationships, find each $k$ by dividing $y$ by its matching nonzero $x$ , make sure the units match, and compare the unit rates instead of the raw numbers. To solve a word problem, check that the situation is proportional, find $k$ , write $y = kx$ and substitute what you know.