Practice Proportional relationships
In a direct proportion, the ratio between corresponding quantities stays constant and the graph passes through the origin.
This models situations such as cost at a fixed unit price.
Exercises with solutions
In a direct proportion, double or triple values belong together: the quotient y ÷ x is always the same. This value is called the constant of proportionality. Calculate via the value for one unit: divide first, then multiply.
Exercise 1
3 exercise books cost €4.50. How much do 5 cost?
Show solution and working
€7.50
First go to one unit: €4.50 ÷ 3 = €1.50. Then times 5: €1.50 · 5 = €7.50.
Exercise 2
Is this relationship directly proportional? A taxi costs €4 base fare plus €2 per kilometre.
Show solution and working
No
In a direct proportion twice the distance would have to cost twice as much. 1 km costs €6, 2 km cost €8 and not €12.
Exercise 3
5 kg of apples cost €12. How much do 2 kg cost?
Show solution and working
€4.80
€12 ÷ 5 = €2.40 for 1 kg. Then €2.40 · 2 = €4.80.
Exercise 4
A machine fills 300 bottles in 4 minutes. How many does it fill in 10 minutes?
Show solution and working
750 bottles
300 ÷ 4 = 75 bottles per minute. Then 75 · 10 = 750.
Common mistake
Building the factor the wrong way round and calculating x ÷ y instead of y ÷ x. Sanity-check the result: more quantity must mean more money. And not every relationship is proportional once a base fee is involved.
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