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 three pencils cost 75p, how much do nine pencils cost?
We know that three lots of three make nine.
So if we multiply the cost of three pencils by 3 we will have the cost of nine pencils:
75p × 3 = 225p
Nine pencils cost £2.25
You may need to use a common factor to find the new quantity.
Question
A recipe for four pancakes uses 300 ml of milk. How much milk is needed for ten pancakes?
Two lots of the recipe will make eight pancakes:
2 × 300 ml = 600 ml
Half of the recipe will make two pancakes:
½ × 300 ml = 150 ml
So to make ten pancakes:
600 + 150 = 750 ml of milk is needed.
Finally, it may be easier to use division to find what one unit is worth, and then multiply to find the final answer.
Example 1
Seven oranges cost £1.75, how much would three oranges cost?
Solution
- One orange would cost £1.75 ÷ 7 = £0.25 = 25p
- So three 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 2
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.
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 five 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 three hours
a) £68 = hourly wage × five 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 three hours’ work is £40.80
