Fractions (Edexcel GCSE Maths): Revision Notes
Fractions
Introduction
Fractions are fundamental mathematical concepts that represent parts of a whole. Understanding how to work with fractions without a calculator is essential for your GCSE mathematics success. This revision note will guide you through all the key methods you need to master.
Simplifying fractions (cancelling down)
When you simplify a fraction, you're making it easier to work with by reducing it to its smallest form. To cancel down or simplify a fraction, you need to divide both the numerator (top number) and denominator (bottom number) by the same number.
The key is to keep going until the top and bottom numbers don't have any common factors. You can cancel down in a series of easy steps, working with any common factors you can spot.
For example, when simplifying , you might first divide both numbers by 2 to get , then divide both by 3 to get the final answer of .
Mixed numbers and improper fractions
Mixed numbers contain both a whole number part and a fraction part (like ). Improper fractions have a numerator that's larger than the denominator (like ). You need to be comfortable converting between these two forms.
Converting mixed numbers to improper fractions:
Think of the mixed number as an addition problem. For example, .
Then turn the whole number into a fraction with the same denominator: .
So .
Converting improper fractions to mixed numbers:
Divide the numerator by the denominator. The answer gives you the whole number part, and the remainder goes on top of the fraction.
For example, remainder , so .
Multiplying fractions
Multiplying fractions is actually quite straightforward once you know the method. You multiply the numerators together and multiply the denominators together.
However, it's often helpful to cancel down first if you can spot common factors. This makes the calculation much easier.
The example shows how to find by cancelling first. You can divide 8 and 12 by their common factor of 4, and divide 15 and 5 by their common factor of 5. This simplifies the calculation significantly before you multiply.
Dividing fractions
Division of fractions follows a simple rule: turn the second fraction upside down and then multiply. This method is sometimes remembered as "keep, change, flip."
When you're working with mixed numbers, always convert them to improper fractions first before attempting division. This makes the calculation much more manageable.
Worked Example: Dividing Mixed Numbers
To find :
Step 1: Convert to improper fractions and
Step 2: Apply "keep, change, flip"
Step 3: Simplify
Common denominators
Finding common denominators is essential for comparing, adding, and subtracting fractions. The easiest method is to find the lowest common multiple (LCM) of all the denominators involved.
This example demonstrates how to order fractions , , and by finding their common denominator of 60. Once all fractions have the same denominator, you can easily compare their numerators to determine the correct order.
Adding and subtracting fractions
Before you can add or subtract fractions, you must ensure they have the same denominator. Once you've found a common denominator, you simply add or subtract the numerators while keeping the denominator the same.
If you're working with mixed numbers, it's often easier to convert them to improper fractions first. This approach helps avoid complications and makes the calculation more straightforward.
The key reminder is that people often find adding and subtracting fractions more challenging than multiplying and dividing, but it becomes much easier once you remember to sort out the denominators first.
Finding fractions of amounts
When you need to find a fraction of a particular amount, you multiply the amount by the numerator and then divide by the denominator. The order doesn't matter - you can divide first if it's easier.
Worked Example: Finding a Fraction of Money
To calculate of £360:
Method 1: Divide first
- £360 ÷ 20 = £18
- £18 × 9 = £162
Method 2: Multiply first
- £360 × 9 = £3240
- £3240 ÷ 20 = £162
Both methods give the same answer of £162.
Practice and application
Understanding fractions requires regular practice with different types of problems. Word problems often involve multiple steps and require you to identify which fraction operations to use.
These practice questions demonstrate the variety of fraction problems you might encounter, from basic calculations with mixed numbers to more complex word problems involving real-world scenarios like baking and cooking.
Key Points to Remember:
- Simplifying fractions: Divide top and bottom by the same number until no common factors remain
- Mixed and improper fractions: Convert between forms by thinking of mixed numbers as addition and using division for the reverse
- Multiplying fractions: Cancel common factors first, then multiply across - it's easier than it looks!
- Dividing fractions: Turn the second fraction upside down and multiply - always convert mixed numbers to improper fractions first
- Adding and subtracting: Get the same denominators first using the LCM, then work with the numerators only
- Finding fractions of amounts: Multiply by the numerator, divide by the denominator (order doesn't matter)