Fractions are a way to show part of a whole. It tells us how many equal parts you have compared to the total number of parts. It is an essential topic in Maths and it is used in everyday situations, whether simplifying, comparing or calculating with fractions.
This article will explain fractions, including how to simplify, compare and multiply with them. We will explore the types of fractions and how to convert fractions into decimals. Quiz questions are included to test your knowledge. This article is suitable for the GCSE Maths syllabus and is aimed at students undertaking GCSE revision. It is suitable for all major exam boards, including AQA and Edexcel.
If you need help with your Fractions GCSE Maths revision, TeachTutti has qualified Maths GCSE tutors who can teach you in person or online. Tutors will tailor lessons to your specific needs, such as running through past paper questions.
What are Fractions?
A fraction is a way to show part of a whole. There are two numbers in a fraction:
- Numerator - this is the top number. It shows how many parts you have of the whole.
- Denominator - this is the bottom number. It shows how many parts the whole is split into.
An example of a fraction is 3/4. The denominator is 4, which tells us the whole is divided into 4 parts. The numerator tells us we have 3 parts of the whole.
Types of fractions
There are different types of fractions. Understanding the variations is important, such as when you need to convert between them or use multiple types in a calculation.
Proper fractions - The numerator is smaller than the denominator. This tells us that value is less than the whole. For example, 3/5 means we have 3 out of 5 parts. 1/4 means we have 1 part out of 8. It is the most common type of fraction in basic calculations.
Improper fractions - The numerator is equal to or greater than the denominator. The value of the fraction is 1 or more. For example, 5/4 means you have 5 quarters, one greater than the whole. 3/3 means you have exactly one whole. You often see improper fractions in multiplication and division. They can be converted into mixed numbers.
Mixed numbers - This combines a whole number and a proper fraction. It’s an alternative to an improper fraction. For example, 1 3/4 means 1 whole and 3 quarters. 2 2/5 means 2 wholes and 2 5ths. It's common to switch between mixed numbers and improper fractions when doing calculations in GCSE Maths, e.g. 1 3/4 = 7/4.
1
Which is an improper fraction?
Simplifying fractions
To simplify a fraction means turning it into the lowest possible terms, while keeping the same value. It's a required step in most exam questions. Simplifying a fraction means writing it in its lowest possible terms, using the smallest numbers that still represent the same value. It’s an expected step in many exam questions, even when not specifically stated. For instance, 12/20 may be correct, but you would need to simplify it to 3/5 for more marks.
You need to divide the numerator and denominator by the highest whole number possible for both. This is called the Highest Common Factor, or HCF:
- 12/20 - The HCF is 4 because it can divide into both. 12 / 4 = 3 and 20 / 4 = 5. This means 12/20 can be simplified to 3/5.
- 36/60 - The HCF is 12. 36 / 12 = 3 and 60 / 12 = 5. This means 36/60 can be simplified to 3/5.
The process is the same even when the original numbers are extremely large or small. In this situation, you may need to simplify it in stages to avoid making mistakes. Start with a safe common number, such as 2, and then continue from there.
Some exam questions will give you a fraction as part of a word problem or a longer calculation. Make sure you keep your workings clear with extended questions, as you will receive higher marks. This skill is also required in algebraic fractions, ratios and when converting between fractions, decimals and percentages.
2
What is the simplified form of 18/24?
Comparing and ordering fractions
You may be asked to compare several fractions, putting them in order of size. It is crucial to place them in the same form at the beginning of your workings. For instance, the fractions 3/5 and 2/3 have different denominations and can't be directly compared.
Start by converting the fractions so they share the same denominator. The easiest approach is to multiply the denominators together. This is called finding a common denominator. Next, you multiply the numerator by the opposite denominator so the fraction doesn't change.
For example:
- Compare 2/3 and 3/5 - The lowest common denominator of 3 and 5 is 15 (3 x 5 = 15). The updated numerator values are 10 (2 x 5) and 9 (3 x 3), which means the fractions are changed to 10/15 and 9/15. This clearly shows that 2/3 > 3/5.
You can also convert fractions into decimals for comparison. This is possible with a calculator or by estimating if the values are simple. It is a quicker method, but can be less precise if the numbers are awkward.
When the denominators match, comparing the fractions becomes easy. The larger numerator is the larger fraction. For instance, 4/7 > 3/7.
If you want to order fractions, convert them so they share a common denominator. Alternatively, you can convert them to decimals. Then, you need to put them into the correct order. The exam question may expect you to convert them back to their original fractional forms.
3
Which fraction is the smallest?
Convert mixed and improper fractions
You need to understand how to convert mixed numbers and improper fractions, particularly in calculations or when solving word problems. First, let's recap these terms:
- Mixed number - this has a whole number and a proper fraction, e.g. 2 1/3.
- Improper fraction - the numerator is equal to or greater than the denominator, e.g. 7/3.
To convert a mixed number into an improper fraction, multiply the whole number by the denominator. Then add the numerator. Make sure you keep the same denominator.
For example:
- Convert 2 1/3 - Multiply the whole number by the denominator (2 × 3 = 6). Add the numerator (6 + 1 = 7). This means 2 1/3 = 7/3.
- Convert 4 5/6 - Multiply the whole number by the denominator (4 × 6 = 24). Add the numerator (24 + 5 = 29). This means 4 5/6 = 29/6.
This conversion is useful when you need to multiply or divide fractions. It's easier to work with one fraction than a combination.
To convert an improper fraction into a mixed number, divide the numerator by the denominator. The result is the whole number, while the remainder is the new numerator.
For example:
- Convert 11/4 - Divide the numerator by the denominator (11 / 4 = 2, with a remainder of 3). This means 11/4 = 2 3/4.
- Convert 22/5 - Divide the numerator by the denominator (22 / 5 = 4, with a remainder of 2). This means 22/5 = 4 2/5.
This is a common process in non-calculator papers, and when measures, money or quantities are involved in the question. You may need to simplify your answer. Always check if the fraction in the mixed number can be reduced.
4
What's 3 2/5 as an improper fraction?
Adding fractions and subtracting fractions
The denominators have to be the same when adding or subtracting fractions. A calculation like 1/4 + 1/3 will always be incorrect, because the pieces are different sizes. The method you use depends on whether the denominators match.
When the denominators match, all you need to do is add/subtract the numerators. For example:
- 3/7 + 2/7 = 5/7
- 6/9 - 4/9 = 2/9
When the denominators differ, you have to find the common denominator before adding or subtracting. When you have finished the calculation, always check if the fraction can be simplified.
For instance:
- Add 1/4 + 1/3 - the lowest common denominator of 4 and 3 is 12 (the first multiples of 4 are 4, 8, 12. The first multiples of 3 are 3, 6, 9, 12). 1/4 becomes 3/12 and 1/3 becomes 4/12. This means 3/12 + 4/12 = 7/12.
- Subtract 5/6 - 1/4 - The lowest common denominator of 6 and 4 is 12 (the first multiples of 6 are 6 and 12. The first multiples of 4 are 4, 8, 12). 5/6 becomes 10/12 and 1/4 becomes 3/12. This means 10/12 - 3/12 = 7/12.
Mixed numbers
When using mixed numbers, the normal practice is to convert them into improper fractions.
For example, 1 2/5 + 3 1/10:
- 1 2/5 = 7/5 and 3 1/10 = 31/10.
- The common denominator for both fractions is 10. 7/5 becomes 14/10. 31/10 doesn't change.
- 14/10 + 31/10 = 45/10.
- This can be simplified to 45/10 = 4 1/2.
You will often see these questions in calculator and non-calculator papers. They are especially common in word problems, where the quantities are shared or combined.
5
What is 1/4 + 2/3?
Multiplying fractions and dividing fractions
Multiplying and dividing fractions doesn't require a common denominator. There are rules to follow for each operation.
Multiplying fractions
To multiply fractions, you need to multiply the numerators together and the denominators together. You can choose to simplify before or after multiplying.
For example:
- 2/3 × 4/5 - Multiply the numerators (2 × 4 = 8) and the denominators (3 × 5 = 15). This means 2/3 × 4/5 = 8/15.
- 5/6 × 3/10 - Multiply the numerators (5 × 3 = 15) and the denominators (6 × 10 = 60). The answer is 15/60, which can be simplified to 1/4.
You can use this method for proper and improper fractions. If you're using mixed numbers, remember to start by converting them into improper fractions:
- 1 1/2 × 2 2/3 - 1 1/2 = 3/2 and 2 2/3 = 8/3. This means 3/2 × 8/3 = 24/6, which can be simplified to 4.
Dividing fractions
You have to flip the second fraction when dividing fractions. This is called finding the reciprocal. Then you multiply. It's a common mistake in calculations, so always make sure you flip the second fraction before multiplying. As always, simplify your answer if possible. For instance:
- 2/3 divided by 4/5 - Flip 4/5 to get 5/4. 2/3 × 5/4 = 10/12 = 5/6.
- 7/10 divided by 2/5 - Flip 2/5 to get 5/2. 7/10 × 5/2 = 35/20 = 7/4 or 1 3/4.
6
What is 2/3 divided by 4/5?
Fractions of amounts
Word problems often ask you to find a fraction of an amount. You will encounter it in ratios, percentages, money and measures. To find the fraction, you have to multiply the amount by the fraction. Remember that "of" means "multiply" in the question. For example, if you're asked to find 3/5 of 40, you're calculating 3/5 x 40.
To find a fraction of an amount, first divide the amount by the denominator. Then multiply the result by the numerator. This will return the required fraction of the whole.
For example:
- Find 2/5 of 30 - Divide the amount by the denominator (30 / 5 = 6) and then multiply by the numerator (6 × 2 = 12). The answer is 12.
- Find 3/4 of 28 - Divide the amount by the denominator (28 / 4 = 7) and then multiply by the numerator (7 × 3 = 21). The answer is 12.
Use a calculator if the numbers don't divide easily. You may see this question in a non-calculator paper, so practice this method for both eventualities.
Mixed numbers
When the fraction is a mixed number, you must convert it first into an improper fraction:
- Find 1 1/2 of 16 - 1 1/2 = 3/2. 16 × 3/2 = (16 divided by 2) × 3 = 8 × 3 = 24.
This type of question often appears in contexts that involve money, recipes, group sizes, and so on. For example, "There are 60 people in a school coach. 2/3 of the seats are taken by students. How many seats are left?" You complete the straightforward calculation 2/3 of 60 = 40. Then subtract this from the whole: 60 - 40 = 20 free seats.
7
What is 3/8 of 64?
Convert fractions to decimals
Converting between fractions and decimals is a common question, such as comparing values, completing tables, or switching between percentage and fraction formats. You may need to find recurring decimals and write them with the correct notation.
The best approach to convert a fraction into a decimal is to divide the numerator by the denominator. You can do this with long division or using a calculator. For example:
- Convert 3/4 into a decimal - 3 divided by 4 = 0.75.
- Convert 1/5 into a decimal - 1 divided by 5 = 0.2.
This process works for both proper and improper fractions. For instance, you can convert 7/4 into a decimal as follows: 7 divided by 4 = 1.75.
Recurring decimals
A recurring decimal is a fraction that repeats forever after the decimal point. A classic example is 1/3, which is equal to 0.3333... and the digit 3 repeats indefinitely. Other examples are:
- 2/9 = 0.222... = 0.2
- 4/11 = 0.363636... = 0.36
Recurring decimals appear in calculator and non-calculator papers. An example question is to spot the recurring decimal in a list, convert it to a fraction, or round it to a specific number of decimal places.
8
What is 5/6 as a decimal?
Conclusion
We see fractions throughout Maths, including basic number work, algebra, ratios and percentages. They are the foundation for many areas of maths.
To improve your ability in fractions, rotate rotating being mixed and improper forms, using operations and converting to decimals. MathsGenie has past paper questions on fractions to test your knowledge. For further reading, SaveMyExams has written an article that explores algebraic fractions.
If you want support learning this or related topics, TeachTutti has a list of top Maths GCSE tutors who can teach you in person or virtually using the TeachTutti learning platform. Every tutor has an enhanced DBS check, and tutors will tailor lessons to your specific needs, such as making revision notes on basic fractions.
