Direct and inverse proportions are complicated topics in the GCSE Maths curriculum. Direction proportion is when two quantities change at the same rate. For example, the price of buying an apple increases directly with the number you purchase. Inverse proportion is when one quantity increases as the other decreases. For example, if more people work on a task, the task will take less time to complete.

These proportions are often seen in exam questions that apply real-life scenarios. This is because we see direct and inverse proportionality in our daily lives, such as to calculate speed, cost and productivity. If you need extra help, TeachTutti has a list of GCSE Maths tutors who can explain this and other topics in the curriculum.

## Direct Proportion

The ratio of two variables remains constant when they are changed in direct proportion. For example, when one variable doubles, the other will as well to maintain the ratio. You often find this relationship in everyday life. The formula for direct proportion is y = kx:

- y is the dependent variable.
- x is the independent variable.
- k is the constant of proportionality.

This means that when x increases or decreases, y will do the same at the rate decided by k.

### Real-life example

If you go shopping and buy apples, you typically see an example of direct proportionality. Let's say that each apple costs 50p. This means the total cost is directly proportional to the number of apples:

- 1 apple costs £0.50.
- 2 apples cost £1.00.
- 3 apples cost £1.50.

The constant of proportionality (k) is 0.50, which is the cost of a single apple. You can express this relationship as: Total Cost = 0.50 × Number of Apples.

### Graphical representation

If you wrote direct proportion as a graph, it would be a straight line passing from the origin (0,0) on a coordinate plane. The slope of this line is determined by the constant of proportionality (k). For instance, if the cost of an apple is shown against the number of apples using the example above, the slope of the line will be 0.50.

### Direct Proportion problems

If you need to answer a problem with direct proportion:

- Find the constant of proportionality (k).
- Use the formula y = kx to find the unknown variable.
- Make sure both variables have the same units.

Example: If 4 pears cost £2.00, what will 7 pears cost?

- Find the constant of proportionality: k = 2.00 / 4 = 0.50.
- Use the formula: Cost = 0.50 × 7 = £3.50.

### 1

If 3 oranges cost £1.20, what is the price of 5 oranges?

## Inverse Proportion

Inverse proportion is when one variable increases and the other decreases while keeping its product constant and unchanged. The general formula is: y = k / x:

- y is the dependent variable.
- x is the independent variable.
- k is the constant of proportionality.

This means that when x increases, y decreases proportionally. The same is true the other way around.

### Real-life example

Let's say there are workers assigned to complete a project. The number of workers decides how long it will take to complete the task: more workers will decrease the time taken to complete the task proportionally:

- If 4 workers can complete a task in 6 hours, the total hours taken for one worker to complete the task is 4 × 6 = 24. This is the constant of proportionality.
- If we increase the number of workers to 6, the time decreases to 4 hours (24 /6).

### Graphical representation

If we draw a graph to represent an inverse proportion, we will see a hyperbolic curve on a coordinate plane. The curve approaches the axes but never touches them. This shows that when one variable keeps increasing, the other variable approaches zero.

### Inverse Proportion problems

Take the following steps to solve a problem with inverse proportion:

- Identify the constant of proportionality (k).
- Use the following formula to find the unknown variable: y = k / x.
- Make sure both variables use the same units.

**Example problem**: 5 works can complete a task in 8 hours. How long will it take 10 workers?

- Find out the constant of proportionality: k = 5 × 8 = 40 hours for one worker to complete the task.
- Use the formula to find the answer: Time = 40 / 10 = 4 hours.

### 2

If 6 people finish a task in 12 hours, how long would the same task take 8 people?

## Solve problems with Proportions

You need to understand the relationship between both variables before you apply the correct formula and solve a problem that involves direct and/or inverse proportions. We have written a few strategies below to help you approach each type of proportion.

### Direct Proportion problems

**Find the Constant of Proportionality**: You need to calculate the constant k first. You will always be able to do this with the information provided in the question. This constant remains the same. For example, if 4 bananas cost £1.20, the constant is k = Total Cost (1.20) / Number of Bananas (4) = 0.30. The constant is that each banana costs £0.30.

**Prepare the equation**: Use the formula y = kx. Here, y is the total cost, x is the number of bananas and k is the constant of proportionality.

**Solve the equation**: After you have written the equation, solve the calculation for the answer. To find the cost of 7 bananas from the above example, y = 0.30 × 7 = £2.10.

### Inverse Proportion problems

**Identify the Constant of Proportionality (k)**: The product of the two variables will be constant for inverse proportion. For instance, it takes 5 workers 8 hours to complete a task. The constant is: k = Number of Workers (5) × Time (8) = 40. Therefore, the time it takes one worker is 40 hours.

**Write your equation**: Use the formula y = k / x where y is the time, x is the number of workers and k is the constant of proportionality.

**Solve the calculation**: Find the answer to your equation to find the variable. If we continue the example above to find out how long 10 workers would take to complete the task: y = 40 / 10 = 4 hours.

### 3

The variable y is directly proportional to x it equals 20 when x = 4. What is the value of y when x = 10?

## Common mistakes

There are several pitfalls that students will often come across using direct and inverse proportions. We have covered the most common below to help you avoid mistakes and improve your understanding of these concepts.

### Example 1: Confusing Direct and Inverse Proportions

Students can confuse the two types of proportion. In direct proportion, both quantities increase or decrease together. Meanwhile, one variable will increase and the other will decrease while keeping the product constant in inverse proportion.

**Example**:

- Direct Proportion: If you work more hours, you earn more money.
- Inverse Proportion: A job takes less time to complete when more people are assigned.

### Example 2: Misapplying formulas

Ensure you use the correct formula - it can be easy to use the formula for the wrong type of proportion. Check if the problem involves direct proportion (y = kx) or inverse (y = k / x) to know which formula to use for your equation.

**Example**:

- Direct Proportion: If the cost of apples is directly proportional to the number of apples, use y=kx.
- Inverse Proportion: If the speed of a car is inversely proportional to travel time, use y = k / x.

### Example 3: Incorrectly calculating the constant

Answers will sometimes miscalculate the constant of proportionality (k). Make sure you find the constant correctly by double-checking before you commit to solving the given problem.

**Example**:

- Direct Proportion: If 5 books cost £20, then each book costs £4 (k = 20 / 5).
- Inverse Proportion: If 3 workers take 12 hours to complete a task, then it takes one worker 36 hours (k = 3 × 12).

### Example 4: Ignoring units

Make sure the units are consistent for both variables. Inconsistent units can lead to mistakes in calculations.

**Example**:

- If the distance is in kilometres and the time is in hours, make sure that the speed is in kilometres per hour rather than metres per second.

### 4

Which scenario is an example of inverse proportion?

## Practice questions and solutions

To master proportions, you need to practice with practice questions, as well as worksheets and GCSE past papers. There are a few practice problems below to help cement your understanding of this topic.

### Direct Proportion example problem

A car rental company charges £15 daily to rent a car. This is directly proportional to how many days you rent the car.

**Problem**: How much does it cost to rent the car for 7 days?

**Solution**:

- Find out the constant of proportionality: k = 15 (cost per day).
- Write your equation using the direct proportion formula: y = 15 × 7.
- Solve your calculation: y = 105. It costs £105 to rent the car for 7 days.

### Inverse Proportion example problem

It takes 8 hours for 5 workers to build a wall. The time it takes is inversely proportional to how many workers there are.

**Problem**: How long will 10 workers take to build the wall?

**Solution**:

- Identify the constant of proportionality: k = 5 × 8 = 40 hours for one worker to build the wall.
- Prepare your equation using the inverse proportion formula: y= 40 / 10.
- Solve your equation: y = 4. It takes 4 hours for 10 workers to build the wall.

### 5

If a house is painted in 9 days by 6 people, how long would it take with just 3 people?

## Practical applications of Proportionality

There are many ways that direct and inverse proportions are used in everyday life. They are also applied in fields as diverse as science, engineering and economics.

### Direct Proportion in everyday life

**Cooking and recipes**:

- The quantities of ingredients in a recipe are directly proportional to the number of servings the food will provide. For example, if a recipe for 4 people requires 200g of flour, a serving for 8 people will need 400g.

**Travel and distance**:

- If you travel at a constant speed, the duration of the journey will be directly proportional to your speed. For example, if you travel at 60 miles per hour, you will cover 120 miles in 2 hours.

### Inverse proportion in everyday life

**Work and time**:

- The time it takes to complete a task is inversely proportional to how many workers are assigned. For example, if 4 workers can finish a task in 10 hours, 8 workers can do it twice the speed in 5 hours.

**Physics and engineering**:

- The light intensity is inversely proportional to the square of the distance from the source. For example, if you move twice as far from a light, the intensity of this light source decreases to one-fourth.

## Conclusion - Proportion revision

You can understand a variety of real-world relationships using direct and inverse proportions, including predicting time and analysing rates. They also help with a variety of maths problems and provide a foundation for more advanced studies.

This is a quick summary of the article:

**Direct Proportion**: Both quantities will increase or decrease together. The formula is y = kx.**Inverse Proportion**: One quantity increases and the other decreases. The formula is y = k / x.**Applications**: These principles can be used widely, from cooking and travel to engineering projects.

If you want to learn more about this area of mathematics, follow the link to learn how to rearrange formulae. You may also want to learn more about probability, which deals with the likelihood of different outcomes and often uses proportional thinking. Follow the link to view CueMath's guide to probability.

If you need further support, consider exploring TeachTutti's list of GCSE Maths tutors to provide personalised guidance.

## Glossary

- Direct proportion - Two quantities increase or decrease together at the same rate. The formula is y = kx.
- Constant of proportionality - This is a constant value (k) that defines the ratio of two directly proportional quantities.
- Inverse proportion - One quantity increases and the other decreases. The formula is y = k / x.
- Dependent variable - This is a variable (y) with a value that depends on another variable in a proportion relationship.
- Independent variable (x) - This variable (x) decides the value of the dependent variable in a proportion relationship.
- Hyperbolic curve - This is a visual representation of an inverse proportion relationship.