If one notebook costs £1.50, then 2 notebooks cost £3, and 10 notebooks cost £15. When you double the number of notebooks, the cost doubles too. When you triple one, the other triples. Two quantities that change together like this are in a proportional relationship.
Let's explore
In a proportional relationship, one quantity is always the same number times the other. That fixed number is called the constant of proportionality, usually written k. If x is the number of notebooks and y is the cost in pounds, then
y=
Watch it explained
What is the Constant of Proportionality? | 7th Grade | Mathcation.com
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Constant of Proportionality Made Easy! Grade 7 Mathematics
4The graph of a proportional relationship is a straight line through the origin (0,0).
k
x
with
k
=
1.5
To find k, divide one quantity by the other:
k=xy
Testing a table. Divide each y by its x. If you get the same answer every time, the relationship is proportional.
x
2
4
6
y
5
10
15
Here 25=2.5, 410=2.5 and 615=2.5. Every ratio is the same, so it is proportional, with k=2.5.
Now compare a taxi that charges £3 to start plus £2 per km. One km costs £5, but 2 km cost £7, not £10. Doubling the distance does not double the price, so this is not proportional.
Testing a graph. A proportional relationship always makes a straight line through the origin, the point (0,0). That makes sense: zero notebooks cost zero pounds. The taxi graph is straight but starts at 3, so it fails the test.
Try it yourself
3 kg of potatoes cost £4.50. Assuming the cost is proportional to the weight, find the cost of 7 kg.
Step 1. Find the constant of proportionality (the price of 1 kg).
k=34.50=1.50
So potatoes cost £1.50 per kg.
Step 2. Write the rule: y=1.5x, where x is kilograms and y is pounds.
Step 3. Substitute x=7.
y=1.5×7=10.5
7 kg cost £10.50.
Graph of cost against weight for potatoes, y = 1.5x: a straight line through the origin, passing through (3, 4.50) and (7, 10.50)
Step 4. Check by scaling: 7 kg is 37 times as much as 3 kg, and 4.50×37=10.50. ✓
Watch out!
Assuming every straight line is proportional. It must also pass through (0,0).
Checking only one pair. One ratio tells you k; you need to check that all pairs give the same value.
Dividing the wrong way. Be consistent: always xy, so k means "y for each 1 of x".
In real life
Fuel at a pump, fabric sold by the metre, and wages paid by the hour are all proportional. Currency exchange works the same way: if £1 buys 1.2 euros, then the constant of proportionality is 1.2, and £50 buys 60 euros.
Which graph shows a proportional relationship?
A curve through the origin
A straight line through the origin
A straight line that starts at 3 on the y-axis
A horizontal line
5 kg of rice costs £12. If cost is proportional to weight, how much do 8 kg cost?