Proportionality Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Proportionality.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A relationship where two quantities maintain a constant ratio: doubling one always doubles the other, giving y=kxy = kx.

If you double one, you double the other. Triple one, triple the other.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Two quantities are proportional when one is always a fixed number times the other.

Common stuck point: The procedure for proportionality is the easy part; the trap is calling any growing pair proportional. Asking "Is y/xy/x the same number for every pair, and is y=0y=0 when x=0x=0?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is y/xy/x the same number for every pair, and is y=0y=0 when x=0x=0?

Worked Examples

Example 1

easy
A car travels 150150 miles in 33 hours at constant speed. How far will it travel in 55 hours?

Answer

The car travels 250250 miles in 55 hours.

First step

1
Find the unit rate (speed): 150 miles3 hours=50\dfrac{150 \text{ miles}}{3 \text{ hours}} = 50 mph.

Full solution

  1. 2
    Distance in 55 hours: 50×5=25050 \times 5 = 250 miles.
  2. 3
    Alternatively, set up a proportion: 1503=d5\dfrac{150}{3} = \dfrac{d}{5}, so d=150×53=250d = \dfrac{150 \times 5}{3} = 250 miles.
Two quantities are proportional when their ratio is constant. Here, distance and time have a constant ratio (speed). Setting up a proportion or multiplying by the unit rate both give the same result.

Example 2

medium
The table shows: x=2,y=8x = 2, y = 8; x=5,y=20x = 5, y = 20; x=9,y=36x = 9, y = 36. Determine whether yy is proportional to xx, and if so write the proportionality equation.

Example 3

medium
A car uses 44 L of fuel per 5050 km. Assuming proportional fuel use, how much fuel does it use for a 325325 km trip?

Example 4

medium
Determine whether each table is proportional. Table A: (x,y)=(1,3),(2,6),(3,9)(x, y) = (1, 3), (2, 6), (3, 9). Table B: (x,y)=(1,4),(2,7),(3,10)(x, y) = (1, 4), (2, 7), (3, 10).

Example 5

medium
A 200200 g serving of cereal contains 2424 g of sugar. How much sugar is in 325325 g of the same cereal, assuming proportionality?

Example 6

hard
The mass mm of a metal cube is proportional to its volume VV. A cube of side 22 cm has mass 7272 g. Find the mass of a cube of side 55 cm of the same metal.

Example 7

hard
yy is proportional to xx. When xx increases from 44 to 99, yy increases by 1515. Find the constant of proportionality.

Example 8

hard
A relationship satisfies y=4x+0y = 4x + 0. Verify it is proportional and state kk. Then check whether y=4x+1y = 4x + 1 is proportional.

Example 9

challenge
At a science fair, the brightness of a bulb is proportional to the cube of the current through it. When I=2I = 2 A, brightness is 4040 units. Find brightness when I=3I = 3 A and verify the proportional model.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
If 1212 apples cost $3.60\$3.60, how much do 2020 apples cost?

Example 2

medium
Is the relationship y=3x+2y = 3x + 2 proportional? Is y=5xy = 5x proportional? Explain.

Example 3

easy
In y=kxy = kx with k=4k = 4, find yy when x=3x = 3.

Example 4

easy
33 apples cost $6\$6. What is the cost per apple?

Example 5

easy
If yy is proportional to xx and y=10y = 10 when x=2x = 2, find kk.

Example 6

easy
Is y=3x+5y = 3x + 5 a proportional relationship?

Example 7

easy
Solve x4=68\frac{x}{4} = \frac{6}{8}.

Example 8

easy
A car travels 6060 km in 11 hour at constant speed. How far in 33 hours?

Example 9

easy
If 44 pens cost $12\$12, what do 66 pens cost at the same rate?

Example 10

easy
A graph passes through the origin and (2,8)(2, 8). What is kk?

Example 11

medium
If 55 workers build a wall in the same time pattern and pay is proportional, 55 workers earn $200\$200. What do 88 earn?

Example 12

medium
On a map, 22 cm represents 55 km. How many km does 77 cm represent?

Example 13

medium
Solve the proportion 3x=915\frac{3}{x} = \frac{9}{15}.

Example 14

medium
A printer prints 2424 pages in 33 minutes. How long for 4040 pages?

Example 15

medium
If y∝xy \propto x and yy increases from 66 to 99, by what factor does xx change?

Example 16

medium
A 33 kg bag of rice costs $7.50\$7.50. At the same rate, what is the cost of 55 kg?

Example 17

medium
Two quantities satisfy ab=cd\frac{a}{b} = \frac{c}{d} with a=6,b=9,c=10a=6, b=9, c=10. Find dd.

Example 18

challenge
If aa is proportional to bb and bb is proportional to cc, show aa is proportional to cc.

Example 19

challenge
Quantity yy is proportional to x2x^2. If y=12y = 12 when x=2x = 2, find yy when x=5x = 5.

Example 20

challenge
Three friends split a cost in the ratio 2:3:52:3:5. If the total is $120\$120, how much does each pay?

Example 21

medium
A recipe needs 33 eggs for 1212 cookies. How many eggs for 2020 cookies?

Example 22

medium
If 77 meters of cloth cost $21\$21, how many meters can you buy with $15\$15?

Example 23

easy
If y=7xy = 7x, find yy when x=5x = 5.

Example 24

easy
If 55 notebooks cost $20, what is the cost of 11 notebook at the same rate?

Example 25

easy
yy is proportional to xx. When x=7x = 7, y=21y = 21. Find kk.

Example 26

easy
At a constant rate, 44 workers paint 22 rooms in 33 hours. How many rooms can 44 workers paint in 66 hours?

Example 27

medium
The cost of cc cans of soup is proportional to cc. If 33 cans cost $4.50, what do 1111 cans cost?

Example 28

medium
yy is proportional to xx. If y=18y = 18 when x=12x = 12, find yy when x=30x = 30.

Example 29

medium
A printer prints 2424 pages in 33 minutes. At the same rate, how long to print 200200 pages?

Example 30

medium
Solve the proportion x18=56\dfrac{x}{18} = \dfrac{5}{6} for xx.

Example 31

medium
On a map, 2.52.5 cm represents 3030 km. How many km does 77 cm represent?

Example 32

hard
In a recipe, the ratio of flour to sugar is 3:23:2. If you use 7.57.5 cups of flour, how much sugar do you need?

Example 33

hard
The graph of a proportional relationship passes through (6,9)(6, 9). Find the value of yy when x=14x = 14.

Example 34

hard
At a constant exchange rate, \$50 USD = €45. How many euros do you get for \$280 USD?

Example 35

hard
Two tanks fill proportionally. Tank A fills 33 L in 44 minutes; Tank B fills 55 L in 44 minutes. How long for Tank A to fill what Tank B fills in 1010 minutes?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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