Proportion – Numerical Problems

Topic Overview

In mathematics, two variables are 'proportional to one another' if a change in one of the variables is always accompanied by a change in the other variable.

There are two types of proportional change: 'direct' and 'inverse'.

When two variables are in 'direct proportion to one another' , they increase or decrease at the same rate.

If variable 'y' is directly proportional to variable 'x' , this can be written as:

\[y \propto x\]

where '∝ ' is the symbol for proportionality.

Alternatively, when two variables are 'inversely proportional to one another' , one variable decreases at the same rate that the other variable increases.

If variable 'y' is inversely proportional to variable 'x', this can be written as:

\[y \propto \frac{1}{x}\]

Key Concepts

In the new linear GCSE Maths paper, you will be required to solve various mathematical problems involving direct and inverse proportions. The specific questions you will be expected to answer will vary depending upon which examination board with which you are registered, but as a rule you will be required to:

  • Understand and use direct proportion
  • Understand and use inverse proportion

Listed below are a series of summaries and worked examples to help you solidify your knowledge about direct and inverse proportions.

Worked Examples

1 - Direct Proportion
During your GCSE maths exam, you will be required to solve various mathematical problems using direct proportion:

Example
(a) - 8 apples costs £4.00. The cost of apples is directly proportional to the number of apples. Using this information, calculate the cost of 24 apples.

Solution
(a) - First, you need to find the cost of one apple. To calculate this cost, divide the total cost of the apples by the number of apples:

(Note: When calculating the cost of items, it is easier to write values such as £4.00 in pennies: i.e. 400p)

400p ÷ 8 = 50p

Therefore, 1 apple costs 50p

Using this value, you can calculate the cost of 24 apples. If one apple costs 50p, the cost of 24 apples is:

50p x 24 = 1200p

(Note: At this point, you can convert the cost from pennies back into pounds) Therefore the cost of 24 apples is £12.00

2 - Inverse Proportion
During your GCSE maths exam, you will also be required to solve various mathematical problems using inverse proportion:

Example
(a) - The time taken to paint a room ( y ) is inversely proportional to the number of people painting the room (x). It takes 3 people 8 hours to paint a room. Using this information, calculate how long it will take 4 people to paint a room.

Solution
(a) - The time taken to paint a room ( y ) is inversely proportional to the number of people painting the room (x). This can be written as :

y ∝ 1/x or y = k/x ; where 'k' is the constant of proportionality.

The question tells us that when y = 8 , x =3 . This can be written as:

8 = k/3

You can now solve this equation to find the value of 'k':

8 = k/3

8 x 3 = k

k = 24

Therefore, the equation which connects the time taken to paint a room and the number of people painting is:

y = 24/ x

You can now substitute your 'x' value into this equation to find the corresponding value of 'y':

When x = 4, y = 24/ 4 = 6

Therefore, it takes 4 people 6 hours to paint a room.

3 - Higher Tier Questions
If you are sitting the Higher Tier GCSE maths exam paper, you may be required to form an equation for two quantities which are either directly proportional to, or inversely proportional to, one another:

Example
(a i) - The cost of DVDs is directly proportional to the number of DVDs. It costs £8.40 for 5 DVDs. Form an equation connecting the cost of DVDs and the number of DVDs.

(a ii) - Use this equation to calculate the cost of 30 DVDs.

Solution
(a i) - As mentioned earlier, if the values 'a' and 'b' are in proportion, this can be written as a ∝ b. If 'a' is the cost of DVDs and 'b' is the number of DVDs (a ∝ b) , you can write this as the equation: a = k x b

Using the values in the question above, you can solve this equation to find the value of 'k' :

840 = k x 5

k = 840 ÷ 5

k = 168

As a result, you can form the equation:

a = 168b

(a ii) Using this equation, you can calculate the price for 30 DVDs by substituting 'b' with your specific value of 30:

a = 168 x 30

a = 5040

Therefore the cost of 30 DVDs is £ 50.40

Example
(b) - The time taken to build a shed is indirectly proportional to the number of people building the shed. It takes 4 people 6 hours to build the shed.

(i) - Form an equation connecting the time ( t ), to the number of people building ( p ).

(ii) - Calculate how long would it take 8 people to build the shed

Solution

(b i) - The variables 't' and 'p' are indirectly proportional to one another. This can be written as:

t ∝ 1/p or t = k x 1/p

By placing the values mentioned in the question into this equation, you can find the value of 'k' :

t = k x 1/p

6 = k x 1/4

6 = k/ 4

k = 24

Therefore the equation connecting 't' and 'p' can be written as:

t = 24/p

(b ii) - Using your equation, you can calculate how long it would take 8 people to build the shed. If you substitute 'p' for your value of '8' :

t = 24/8

t = 3

Therefore, it would take 8 people 3 hours to build the shed.

Exam Tips

  1. It is important to remember the difference between direct and indirect proportion. When two variables are in 'direct proportion to one another' they increase or decrease at the same rate.
  2. Alternatively, when two variables are 'inversely proportional to one another' one variable decreases at the same rate that the other variable increases.
  3. Remember that, if variable 'y' is directly proportional to variable 'x' , this can be written as: y ∝ x or y = kx.
  4. Remember that, if variable 'y' is inversely proportional to variable 'x', this can be written as: y ∝ 1/x or xy = k.

Topic Summary

Once you understand the fundamental differences between direct and inverse proportion, you with be able to tackle numerical proportion questions with confidence and ease! If you memorise the processes for solving direct and inverse proportion problems respectively, you can apply these steps to any examination question. In this manner, it is incredibly useful to write down all of your working out. Not only will you earn maximum method marks, but you will also be able to double check your working and ensure that you have completed all of the necessary processes correctly.

Related Topics

  • Repeated Proportional Change
  • Proportion: Algebraic Problems
  • Problem Solving with Decimals and Percentages
  • Ratio Problems
  • Compound Measure Problems
  • Pre-calculus Skills