### GCSE Maths

Numbers
Algebra
Geometry and Measures
Probability
Statistics

# Direct Proportion

Direct proportion is a mathematical concept that describes the relationship between two variables when they increase or decrease at the same rate. In other words, when one variable increases, the other variable also increases, and vice versa.

Direct proportions are when two values increase or decrease together by the same factor. For example, there is a direct proportion between centimetres (cm) and metres (m), as 100 cm equals 1 m, so the multiplier is always 100.

The proportionality symbol is:

• It is used to indicate that one expression changes in proportion to another

We can use it to represent a directly proportional relationship. For example, if is directly proportional to p, we can write .

## Constant of Proportionality (k)

In a directly proportional relationship, there is a constant of proportionality, denoted as . This constant represents the ratio between the two variables, expressed as .

The constant can be calculated from the information provided about x and y. If y is directly proportional to x, we write y ∝ x, which is equal to . If t is directly proportionate to , then this means that , where is a natural number showing that is a multiple of .

## Solving Direct Proportion Problems

To solve direct proportion problems, follow these four steps:

Step 1: Find the proportional relationship between the variables: Identify the variables that are directly proportional and write the relationship using the proportionality symbol (∝).

Step 2: Form an equation with the constant of proportionality: Replace the proportionality symbol with an equal sign and include the constant of proportionality (k) in the equation, e.g., .

Step 3: Use the information provided to find the constant of proportionality: Using the values provided for x and y, calculate the constant of proportionality (k) by dividing one variable by the other, e.g., .

Step 4: Substitute the constant of proportionality into the equation: Once you have calculated the constant of proportionality ( ), substitute it back into the equation to find the unknown values.

Let’s look at an example:

If and when , find the value of when .

First, write the proportion relationship: .

Convert to an equation using the constant of proportionality: .

Use the given information to find the constant of proportionality:   Substitute the constant of proportionality into the equation: .

## Direct Proportion in Graphs

Direct proportion graphs have specific characteristics. They are straight lines that pass through the origin (0,0) and have a constant slope equal to the constant of proportionality, .

• This graph has the equation To plot a direct proportion graph, first calculate the constant of proportionality and create a table of values. Then, plot the points on the graph and draw a straight line through the points and the origin.

Let’s look at an example:

If and when , plot the graph of the direct proportion.

First, calculate the constant of proportionality: .

Create a table of values, such as:

Plot these points on the graph and draw a straight line through the points and the origin (0,0).

## Example

Example 1:

Suppose the cost of apples is directly proportional to the number of apples bought. If 4 apples cost £12, how much would 10 apples cost?

1. Write the proportion relationship: Cost ∝ Number of apples

2. Convert to an equation using a constant of proportionality: Cost Number of apples

3. Use given information to find the constant of proportionality:  4. Substitute the constant of proportionality into the equation: Cost = 3 × Number of apples

5. Calculate the cost for 10 apples: Cost £ 10 apples would cost £30.

Example 2:

Suppose that y varies directly as and when .

Form an equation connecting x and y, then use it to find the value of y when x=5. Also, find the value of x when y=108.

We are given that is directly proportional to the square of , which can be expressed as . To form an equation connecting and , we introduce a constant of proportionality, , resulting in the equation .

To find the value of , we use the given data point: when , .

Substitute the values into the equation:   So, the equation connecting and is .

Now, we can use this equation to find the value of when :   When , .

Next, we’ll find the value of when :     When , can be either or .

Example 3:

The speed of a car is directly proportional to the force applied to its accelerator pedal. If applying a force of 20 N results in a speed of 40 km/h, what would be the speed when a force of 50 N is applied?

1. Write the proportion relationship: Speed ∝ Force

2. Convert to an equation using a constant of proportionality: Speed Force

3. Use given information to find the constant of proportionality:  4. Substitute the constant of proportionality into the equation: Speed Force

5. Calculate the speed for a force of 50 N: Speed 100 km/h

The speed would be 100 km/h when a force of 50 N is applied.