Direct proportion - Intermediate and Higher tier
Two quantities are said to be in direct proportion if they increase or decrease in the same ratio.
If two amounts are directly proportional we can scale the quantities up by multiplying.
Question
If 3 pencils cost 75p, how much do 9 cost?
We know that 3 lots of 3 make 9.
So if we multiply the cost of 3 pencils by 3 we will have the cost of 9 pencils:
75p × 3 = 225p
9 pencils cost £2.25
You may need to use a common factor to find the new quantity.
Question
A recipe for 4 pancakes uses 300 ml of milk. How much milk is needed for 10 pancakes?
We are going from 4 pancakes to 10 pancakes.
A common factor of 4 and 10 is 2.
2 pancakes will need half the amount of milk:
½ of 300 ml = 150 ml
To get from 2 pancakes to 10 pancakes we multiply by 5:
150 ml × 5 = 750 ml of milk is needed.
Finally, it may be easier to use division to find what 1 unit is worth, and then multiply to find the final answer.
Example one
7 oranges cost £1.75, how much would 3 oranges cost?
Solution
- 1 orange would cost £1.75 ÷ 7 = £0.25 = 25p
- So 3 oranges would cost 25p × 3 = 75p
The symbol for direct proportion is ∝.
A ∝ B means A is directly proportional to B.
This can be converted into a formula:
\(\text{A = k × B}\)
where \(\text{k}\) is the constant of proportionality. It shows the relationship between A and B.
Example two
A babysitter is paid £6.25 for each hour of work. Here, earnings are directly proportional to the number of hours worked. We can write this as:
earnings ∝ hours worked
We can create a formula to show the relationship between these two values: earnings = £6.25 × hours worked.
Here the earnings per hour of £6.25 is the constant of proportionality.
This formula makes it easy to calculate earnings or the number of hours worked given the other value.
Question
A builder’s pay is directly proportional to the number of hours worked. For 5 hours’ work a builder’s pay is £68.
a) Write a formula to show the relationship between wages and number of hours worked and use this to calculate the constant of proportionality
b) Use this formula to calculate a builder’s wage given they have worked 3 hours
a) £68 = hourly wage × 5 hours.
£68 ÷ 5 hours = hourly wage.
Hourly wage = £13.60 per hour.
Formula is wages = £13.60 × hours worked.
b) Wages = £13.60 × 3 = £40.80
The builder’s wage for 3 hours’ work is £40.80
