Topic 1 of 7

Proportional relationships

Step 9 · Proportional reasoning topics

Recognize when two amounts are proportional, find the constant of proportionality from a story, a table or a graph, and write it as an equation y = kx.

Constant of proportionality

Two amounts are proportional when one is always the same number times the other. That number is the constant of proportionality, usually called k. It is the unit rate: how much for one.

Example: Finding k

3 pounds of apples cost $6. What is the cost per pound?

Divide the cost by the pounds: 6 ÷ 3 = 2.
Every pound costs $2, so k = 2.

Answer$2 per pound

k can come from fractions too. If you walk $\frac{1}{2}$ mile every $\frac{1}{4}$ hour, your speed is $\frac{1}{2}$ ÷ $\frac{1}{4}$ .

Example: A unit rate with fractions

Sam walks $\frac{1}{2}$ mile every $\frac{1}{4}$ hour. How far does he walk in one hour?

One hour holds 4 quarter hours.
4 × $\frac{1}{2}$ = 2.

Answer2 miles per hour

Tables and graphs

In a table of a proportional relationship, y ÷ x is the same in every row. On a graph, the points lie on a straight line that goes through the origin, (0, 0).

Pens, x	Cost, y	y ÷ x
1	$3	3
2	$6	3
4	$12	3

Every row gives the same y ÷ x: 3 dollars a pen.

Pens at $3 each: the points lie on a line through (0, 0). The point (1, 3) shows k = 3.

On the graph, the point where x is 1 sits at height k: it is the point (1, k).

Equations y = kx

Every proportional relationship can be written y = kx. Knowing k lets you find y for any x, and x for any y.

Example: Writing the equation

Gas costs $4 a gallon. Write an equation for the cost y of x gallons, and find the cost of 7.5 gallons.

k = 4, so y = 4x.
y = 4 × 7.5 = 30.

Answery = 4x, and 7.5 gallons cost $30

Example: Working backwards

With y = 4x, how many gallons cost $50?

50 = 4x.
x = 50 ÷ 4 = 12.5.

Answer12.5 gallons

Try it yourself

Work each one on paper first, then open the answer to check.

5 notebooks cost $15. What is the constant of proportionality, in dollars per notebook?
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3
A table has x = 4, y = 10 and x = 6, y = 15. Is it proportional, and what is k?
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Yes. 10 ÷ 4 = 2.5 and 15 ÷ 6 = 2.5, so k = 2.5.
A line through (0, 0) passes through (1, 7). What equation does it show?
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y = 7x
With y = 6x, what is y when x = 3.5?
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21
With y = 6x, what is x when y = 45?
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7.5
Ana reads $\frac{1}{3}$ of a chapter every $\frac{1}{6}$ hour. How many chapters is that an hour?
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2 chapters an hour: $\frac{1}{3}$ ÷ $\frac{1}{6}$ = 2.