Math Core

Some quantities grow together in a perfectly steady way: buy twice as many tickets and you pay twice as much. Others don't: a taxi ride twice as long usually costs less than twice as much, because of the starting fee. In this lesson you'll learn how to tell the difference.

Same ratio every time

Here is a table for cooking rice. The cups of water depend on the cups of rice.

cups of rice, $x$	cups of water, $y$	$\dfrac{y}{x}$
2	3	$\dfrac{3}{2} = 1.5$
4	6	$\dfrac{6}{4} = 1.5$
6	9	$\dfrac{9}{6} = 1.5$
10	15	$\dfrac{15}{10} = 1.5$

Every row has the same ratio: 1.5 cups of water for each cup of rice. All the ratios $2:3$ , $4:6$ , $6:9$ and $10:15$ are equivalent. That's what makes this relationship special.

Definition

Proportional relationship

Two quantities $x$ and $y$ are in a proportional relationship if the ratio $\dfrac{y}{x}$ is the same for every pair of values. In other words, you always multiply $x$ by the same number to get $y$ .

How to check a table

Test for a proportional relationship

For each row (with $x \ne 0$ ), divide: $\dfrac{y}{x}$ .
If every quotient is the same, the relationship is proportional.
If even one quotient is different, it is not proportional.

If the table includes $x = 0$ , a proportional relationship must have $y = 0$ there too. Zero rice needs zero water.

Worked example: A gym membership

A gym charges this much for a membership:

months, $x$	total cost in dollars, $y$
1	35
2	60
3	85

Is the cost proportional to the number of months?

Divide in each row:

\frac{35}{1} = 35, \qquad \frac{60}{2} = 30, \qquad \frac{85}{3} \approx 28.33

The ratios are not equal, so the relationship is not proportional. (The gym likely charges a $10 sign-up fee plus $25 a month. That extra fee breaks the pattern.)

Common mistake

Don't check only the first two rows. A table can look proportional at first and then break later. Check every row before you decide.

Checking a situation without a table

You can also reason about a situation directly. Ask: "If I double one quantity, does the other quantity exactly double? If one is zero, is the other zero?"

Worked example: Which situations are proportional?

The perimeter of a square and its side length.
The area of a square and its side length.
Your age and your little brother's age, if he is 3 years younger.

Solutions.

Proportional. Perimeter $= 4 \times \text{side}$ every time. Side 2 gives 8, side 5 gives 20. The ratio is always 4.
Not proportional. Side 2 gives area 4 (ratio 2), but side 3 gives area 9 (ratio 3). The ratios change.
Not proportional. When you're 10, he's 7 (ratio 0.7). When you're 20, he's 17 (ratio 0.85). You add 3 each time; you don't multiply by the same number.

Tip

A relationship that uses "add the same amount" (like a fixed fee or an age difference) is almost never proportional. A relationship that uses "multiply by the same amount" (like a price per item) is.

Using a proportional relationship

Once you know a relationship is proportional, you can find missing values, because every ratio is the same.

Worked example: Finding a missing value

The number of pages Priya reads is proportional to the time she spends reading. She reads 12 pages in 5 minutes. How many pages does she read in 15 minutes?

Method 1: scale up. 15 minutes is 3 times as long as 5 minutes, so she reads $3 \times 12 = 36$ pages.

Method 2: use the ratio. The ratio is $\dfrac{12}{5} = 2.4$ pages per minute. In 15 minutes she reads $2.4 \times 15 = 36$ pages.

Both methods give 36 pages.

Practice

Practice 1

Which table shows a proportional relationship?

Practice 2

The table shows a proportional relationship. What number goes in the blank?

$x$	$y$
3	12
5	20
8	?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

Which situation is not a proportional relationship?

Practice 4

Is this relationship proportional?

$x$	4	6	10
$y$	10	15	24

Practice 5

A kayak rental costs $20 plus $8 per hour. Is the total cost proportional to the number of hours?

Practice 6

The cost of pencils is proportional to the number of pencils. 9 pencils cost $2.25. How many dollars do 15 pencils cost?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

The table shows a proportional relationship. What is the missing value of $x$ ?

$x$	$y$
2	5
7	17.5
?	30

Enter a number. Fractions like 3/4 and sqrt(2) are OK.