Ratios are everywhere in our day-to-day life. They play an important role in various decisions, whether we're cooking a meal or deciding our revision schedule against time to relax. Getting the hang of ratios will give you a tool to help solve everyday problems.
This article explains ratios and how to calculate and simplify them. The examples given show their use in different scenarios. Quizzes are included at the end of each section to test your knowledge. If you need further support, TeachTutti has a list of qualified Maths GCSE tutors to help you revise this topic.
Understanding ratios: GCSE curriculum
Ratios are a core concept in maths. Put another way, you must understand ratios to grasp more complex topics. It is particularly useful in statistics, geometry, science and everyday problem-solving.
A ratio is a way to compare two or more quantities, showing how many times one quantity is larger or smaller than another. It is important for decision-making and problem-solving because it provides a way to quantify relationships and making meaningful comparisons.
We can express a ratio in three ways:
- As a fraction: If there are 4 bananas and 8 apples, we can write the ratio of bananas to apples as 1/2. This means there are two times as many apples compared to bananas.
- With a colon: Using the same fruit example, this ratio can also be expressed as 1:2.
- Using the word 'to': You could also see this comparison written as "1 to 2". This is the least common way to express a ratio.
Types of ratios
There are different types of ratios that you will come across:
- Part-to-part: This ratio compares different parts of a group to each other, like the apples to oranges above. Both of these items are in the same group of fruit.
- Part-to-whole: This compares one part of the group to the entire group. In the fruit example, there are 12 pieces of fruit (4 bananas and 8 apples). There are 4 bananas with a part-to-whole ratio for all the fruit of 4:12. This simplifies to 1:3.
- Rates: A rate is a unique ratio where the quantities being compared have different units. Taking the fruit example further, this could be carbohydrates or calories for bananas and apples.
You will often see pie charts, bar models and other illustrations to make ratios easier to understand. Seeing how much one quantity occupies a pie chart compared to another is a useful representation of their relationship. These visual aids are common in sports, such as showing the possession statistics between two teams in a football match.
1
You're preparing a fruit salad that needs a ratio of apples to oranges of 2:3. If you already have 6 apples, how many oranges do you need?
Calculate a ratio
Now we understand the basics of ratios, we need to learn how to calculate and simplify them. For example, 1:2 is far easier to understand than an un-simplified ratio of 56:112.
Calculating ratios means finding the relationship between two or more quantities. This is fairly straightforward with numbers but can be more complicated if we need to use another unit of measurement. For example, we may need to convert kilograms to litres or millilitres first before finding the ratio.
Example:
A school project is to create a scale model of a bridge. The actual bridge is 500 meters long and 40 meters wide. Your scale model needs to be 100 times smaller. The ratio of the model’s length to width needs to be the same as the actual bridge.
- Length of the model - To find the model length, we need to divide the actual length by the scale factor. This is 500 / 100 = 5 meters
- Width of the model = To find the model width, we need to divide the actual width by the scale factor. This is 40 / 100 = 0.4 meters
The ratio of length to width remains 5:0.4. We can simplify this to 12.5:1.
Simplifying ratios
To simplify a ratio means to reduce the numbers in the ratio to their smallest form while keeping the same relationship. You simplify ratios by dividing each quantity by their shared greatest common divisor (GCD).
- Find the GCD: Look for the highest number that divides both parts of the ratio, keeping each part a whole decimal without a remainder.
- Divide: Divide each part of the ratio by the GCD that you have found.
Example:
If you have a ratio of 18:12, the greatest common divisor of 18 and 12 is 6. If we divide 18 and 12 by the GCD, we are left with 3:2.
This simpler form makes it easier to see that for every 3 units of one quantity, there are 2 units of another.
2
You need to landscape a garden and want a ratio of flowering plants to green plants of 3:2. If there are 45 flowering plants, how many green plants do you need?
Use ratios in problem-solving
Ratios give us a precise way to scale quantities up or down while keeping the same proportions. This is essential in areas that include:
- Recipes: You can change the quantity of each ingredient to make a larger or smaller dish while keeping the taste and texture the same.
- Chemistry: To create the desired chemical reaction, you need to mix chemicals with precise ratios to avoid wastage or dangerous outcomes.
- Finance: Ratios are used to analyse financial statements. For example, a ratio called "debt-to-equity" is used to discover the balance between money borrowed by a business and the funds owned by shareholders.
When we have a question that requires the use of ratios, we need to be sure what the ratio is comparing and use the right calculation method to find the solution.
Example:
A landscape architect is designing a park and needs to maintain a grass-to-paved area ratio of 3:1 so there's enough green space for the public. If the total area of the park is 16,000 square meters, how much of it should be paved?
- The ratio is 3 parts grass and 1 part paved. This means a total of 4 parts.
- Each part is the total (16,000) divided by 4 = 4,000 square meters.
- This means that the paved area (1 part) will be 4,000 square meters.
This calculation keeps the ratio properly maintained and meets the design requirements for the park.
3
In a classroom, the ratio of students to computers is 3:1. If there are 36 students in the class, how many computers are needed?
Common mistakes and tips
There are common mistakes to watch for when using ratios. One frequent error is the misalignment of units or mismatched terms. For example, if we compare weights to volumes without converting the units correctly, it will lead to incorrect conclusions. Always make sure the units for both products are consistent and have converted if needed.
You also need to watch you simplify the ratios correctly. Students sometimes try to simplify a ratio without using the greatest common divisor, which can lead to a partially simplified ratio. This makes it more challenging to work with the simplified ratio and leads to follow-up errors when resolving a multi-step problem.
Tips for success
To use ratios accurately, use the following tips in your calculations:
- Practice regularly: Irritating as it may sound, practice really does make perfect. Work through from basic to increasingly complex ratios to increase your understanding and confidence.
- Visualise the problem: Draw a diagram or even use a physical object to help you see the relationship shown by the ratio. It's an especially useful technique when you're dealing with complex scenarios.
- Check your work: Always double-check your calculations and the logic of your ratios. It's a good idea to invert the operation and see if you return to the original quantities.
4
You need to mix paint to create a custom colour. The ratio of red to blue paint is 3:2. If you start with 6 litres of red paint, how many litres of blue paint do you need?
Ratios in advanced applications
Ratios are used extensively in advanced fields including statistics, engineering and economics.
Statistics
Ratios are used in statistics to better understand relationships between different data sets. For example, the Odds ratio in medical studies compares the likelihood of an event happening in two groups. It's crucial to determine the effectiveness of a treatment or the risk factor associated with a medical condition.
Engineering
Structures are built securely and functionality in no small part due to the use of ratios in engineering:
- The Gear ratios in machines determine the speed and torque of moving parts, which are essential for designing efficient mechanical systems.
- Aspect ratios are used in civil engineering to influence the stability and aesthetics of buildings.
Economics
Economists use ratios to check the financial health and market conditions of a business. There are important economic ratios like "price-to-earnings" that investors use to assess the value of a company before deciding to buy or sell stocks. Liquidity ratios are also important in calculating if a company can cover short-term liabilities using its current assets.
Applying advanced ratios
We can solve complex problems and understand broader applications by learning to use ratios effectively. Whether you're analysing data, designing a machine, or evaluating investment opportunities, a strong foundation in ratios allows for better decision-making and solutions.
5
An investor analyses two companies; Company A has a price-to-earnings ratio of 25, while Company B has a ratio of 15. If all other factors are equal, which company's stock represents a better value for money?
Conclusion
Ratios are used in various fields and extend well beyond the GCSE curriculum. Their use is as diverse as finance, where they aid in investment decisions, to everyday tasks like cooking or project planning. Getting the hang of ratios can simplify decision-making and improve our problem-solving ability.
Use Khan Academy to test your knowledge with online worksheets of basic and advanced ratios. For students or parents seeking 1-2-1 guidance in mathematics, consider exploring available tutors who specialise in GCSE Maths at TeachTutti, to help you improve your understanding of this and other topics in the GCSE curriculum.
Frequently asked questions
A ratio is a comparison of two quantities - it tells us how much there is of one quantity compared to another e.g. 2:6. A proportion is an equation that shows two ratios are equal. It's used to find an unknown quantity within a group of ratios, such as 3:2 = 6:x.
To convert a ratio into a decimal, you need to treat the ratio as a fraction. We can write the ratio 3:4 as 3/4. We then divide the numerator by the denominator: 3 ÷ 4 = 0.75.
You can set up a proportion based on a known ratio and use this to find an unknown quantity. If you know the ratio of students to teachers is 20:1 and there are 100 students, you can make a proportion to find out that there would be 5 teachers.
Ratios are used in many daily activities:
- Cooking: Adjusting the quantities of ingredients in a recipe.
- Construction: Mixing materials like cement and sand in precise proportions to create the desired product.
- Finance: Analysing financial ratios to look at the health of a business.
- Medicine: Calculating dosages for medications based on weight.
Ratios are used to understand maps and models among other scales. A scale such as 1:100 means that one unit of measurement represents 100 of the same units. This helps to convert measurements into real-world distances in a map for example.
- Find out the greatest common divisor (GCD) of the quantities in the ratio.
- Divide each number by the GCD in the ratio.
- Ensure that the ratio is now in its simplest form.
Glossary
- Ratio - A comparison of 2+ quantities, showing how many times one value is greater/smaller than the other.
- Proportion - An equation that states two ratios are equal. It helps for solving problems where one part of the ratio is unknown.
- Rate - A specific kind of ratio where the quantities have different units, like speed (miles per hour) or density (people per square mile).
- Scale
- The ratio of the dimensions of a model to the dimensions of the original. This is often used in drawings, models and maps. - Scale factor - A number which scales or multiplies some quantity. It is used to adjust the sizes or amounts in direct proportion to the factor.
- Direct proportion - When two quantities increase or decrease at the same rate. This means their ratio remains constant.
- Inverse proportion - A relationship where one quantity increases as another decreases.
- Simplify - Reducing a ratio to its smallest form by dividing both terms by their greatest common divisor.
- Part-to-part - A comparison between two distinct groups in a larger set, focusing on the relationship between these two groups alone.
- Part-to-whole - A comparison where one part is compared to the total, combining all parts into a total.
This post was updated on 30 Nov, -0001.
