Proportional relationships

Two amounts are in a proportional relationship when their ratio never changes. The table has a constant ratio and the graph is a line through 0.

The idea: a ratio that stays the same

Apples cost $2 a pound. 1 pound costs $2, 3 pounds cost $6 and 5 pounds cost $10. Cost ÷ pounds is always 2, so cost is proportional to weight.

Now a taxi charges $3 to start plus $2 a mile. 1 mile costs $5 and 2 miles cost $7. The ratios are 5 and 3.5, so this relationship is not proportional.

See it: pictures to draw with your child

Copy these onto paper or a whiteboard. A quick sketch makes the idea concrete before any written method.

01234560246810121416poundsdollarsy = 2x
Apples at $2 a pound: a straight line through (0, 0), so proportional.
01234560246810121416milesdollarsy = 2x + 3
A taxi with a $3 start: a straight line that misses (0, 0), so not proportional.
poundsdollars1236510
In the table, dollars ÷ pounds is always 2.

Test 1: check the table

Divide each y by its x. If every answer is the same, the relationship is proportional. That shared number is the constant of proportionality, k.

A table that includes (0, 0) is a good sign, but you still need to check the ratios. Zero cost for zero pounds fits; $3 for zero miles does not.

Test 2: check the graph

A proportional relationship makes a straight line through the origin, the point (0, 0). Both things have to be true.

A straight line that crosses the y-axis above 0 is not proportional, like the taxi fare. A curved line is not proportional either, even if it starts at 0.

Test 3: check the equation

A proportional relationship can be written y = kx, where k is a number. Cost = 2 × pounds is proportional.

An equation with something added, like y = 2x + 3, is not proportional. Neither is y = x², because the ratio y ÷ x keeps changing.

Check two pairs

Type two (x : y) pairs from a table. If the ratios are equivalent, the pairs could be proportional.

Proportional relationships in real life

Many everyday rates are proportional: pay at a fixed hourly rate, the cost of fuel by the litre and the distance at a steady speed.

Others are not. A phone plan with a monthly fee, a taxi with a starting charge and a child’s height against age all fail the tests. Asking “is it proportional?” is a great habit.

Quiz: proportional or not?

Decide from the table or equation, or find k.

Where this fits: ratios step by step

  1. What is a ratio
  2. Part-to-part and part-to-whole ratios
  3. Equivalent ratios
  4. Simplifying ratios
  5. Ratio tables
  6. Double number lines
  7. Tape diagrams
  8. Comparing ratios
  9. Unit rates
  10. Unit price and best buy
  11. Speed, distance and time
  12. Solving proportions
  13. Proportional relationships (this page)
  14. Constant of proportionality
  15. Scale drawings and maps
  16. Ratio word problems

What to say to your child

A short script for teaching it the first time. Pause after each line and let your child do the next step.

  1. “Apples cost $2 a pound. What do 3 pounds cost?” (6 dollars.)
  2. “Divide the cost by the pounds. What do you get each time?” (2.)
  3. “The taxi costs $5 for 1 mile and $7 for 2 miles. Divide. Same answer?” (No: 5 and 3.5.)
  4. “Where does each graph start?” (The apples at 0, the taxi at 3.)
  5. “So which one is proportional?” (The apples.)

More worked examples

Example 1: a table

x: 2, 4, 6. y: 5, 10, 15. y ÷ x = 2.5 every time, so it is proportional, with k = 2.5.

Example 2: a table that fails

x: 1, 2, 3. y: 4, 7, 10. y ÷ x = 4, 3.5, 3.33. Not proportional.

Example 3: an equation

y = 7x is proportional with k = 7. y = 7x + 1 is not.

Example 4: a story

A pool fills at 15 litres a minute. Litres = 15 × minutes, so it is proportional.

Practice questions (answers below)

Ask these out loud or write them down. Check each answer, and ask “how did you get that?” for one or two.

  1. x: 1, 2, 3. y: 3, 6, 9. Proportional?
  2. x: 1, 2, 3. y: 2, 4, 7. Proportional?
  3. y = 5x. Proportional?
  4. y = x + 5. Proportional?
  5. A line through (0, 0) and (4, 8). Proportional?
  6. A line through (0, 2) and (2, 6). Proportional?
  7. $12 an hour pay. Proportional?
  8. $10 a month plus $1 a song. Proportional?
  9. x: 2, 5, 8. y: 1, 2.5, 4. k?
  10. Stretch: is y = 3/x proportional?
Show the answers
  1. Yes, k = 3
  2. No
  3. Yes
  4. No
  5. Yes, k = 2
  6. No
  7. Yes
  8. No
  9. 0.5
  10. No, the ratio y ÷ x changes

Common mistakes

Any straight line countsCalls y = 2x + 3 proportional because it is a straight line.Fix: the line must also pass through (0, 0).
Checking only two pointsFinds the same ratio for two rows and stops.Fix: check every row of the table.
Dividing x by yWorks out x ÷ y and gets a different k.Fix: k is y ÷ x. Be consistent and it still works, but k means y per x.

How to teach it at home

Shop receipt

Check if price is proportional to amount for loose fruit.

Taxi talk

Work out a taxi fare with a starting charge and see why it is not proportional.

Walk graph

Plot distance walked against minutes and look for a line through 0.

By grade

More for each grade: Grade 7 math and the CBSE class 7 maths guide.

Worksheets and practice

Proportional relationships worksheet, page 1

Proportional relationships worksheet: printable PDF with an answer key. · Make a fresh test

Ratio practice · ratio calculator · unit rate calculator · ratio games · ratios for kids · percentages

Last reviewed

Questions parents ask

What is a proportional relationship?

A relationship where y ÷ x is always the same number, so y = kx.

How do you know if a table is proportional?

Divide each y by its x. If every answer is the same, the table is proportional.

How do you know if a graph is proportional?

It is a straight line through the origin, (0, 0).

Is y = 2x + 3 proportional?

No. It is a straight line, but it does not pass through (0, 0).

What grade learns proportional relationships?

Grade 7 under Common Core 7.RP.A.2. In India, direct proportion in class 8.