Fraction Operations
Master adding, subtracting, multiplying and dividing fractions — essential skills used in everyday maths and beyond.
Adding & Subtracting Fractions
To add or subtract fractions, both fractions must have the same denominator (called the common denominator). Once the denominators match, simply add or subtract the numerators and keep the denominator.
Step-by-Step: Adding Unlike Fractions
Find the LCD
Lowest Common Denominator of 3 and 4 is 12
Convert both
1⁄3 = 4⁄12 1⁄4 = 3⁄12
Add & simplify
4⁄12 + 3⁄12 = 7⁄12
Same denominators
3⁄7 + 2⁄7 = 5⁄7
Just add the numerators!
Different denominators
2⁄3 − 1⁄4 = 8⁄12 − 3⁄12 = 5⁄12
Find the LCD first.
Multiplying & Dividing Fractions
Multiplying Fractions
Multiply straight across — numerator × numerator, denominator × denominator. No need to find a common denominator!
2⁄3 × 3⁄5 = 6⁄15 = 2⁄5
Simplify by dividing by the HCF (3)
Dividing Fractions
Keep the first fraction, change the division sign to multiplication, and flip (reciprocal) the second fraction. Known as "Keep, Change, Flip".
3⁄4 ÷ 2⁄5 = 3⁄4 × 5⁄2 = 15⁄8 = 17⁄8
Flip 2⁄5 to get 5⁄2
Remember: Always simplify your final answer!
Check if the numerator and denominator share a common factor. If they do, divide both by that factor. For example, 6⁄8 simplifies to 3⁄4 (both divided by 2).
Working with Mixed Numbers
A mixed number has a whole number part and a fraction part (e.g., 2½). Before operating, convert to an improper fraction where the numerator is larger than the denominator.
Converting: Mixed Number → Improper Fraction
Key Vocabulary
Numerator
The top number of a fraction — it shows how many parts we have.
Denominator
The bottom number of a fraction — it shows how many equal parts the whole is divided into.
Lowest Common Denominator (LCD)
The smallest number that is a multiple of both denominators. Used when adding or subtracting unlike fractions.
Reciprocal
The flipped fraction. The reciprocal of 3⁄4 is 4⁄3. Used when dividing fractions.
Worked Examples
Calculate 2⁄5 + 1⁄3
Step 1: LCD of 5 and 3 is 15.
Step 2: Convert: 2⁄5 = 6⁄15 and 1⁄3 = 5⁄15
Step 3: Add numerators: 6⁄15 + 5⁄15 = 11⁄15
Answer: 11⁄15 (already in simplest form)
Calculate 4⁄5 × 5⁄8
Step 1: Multiply numerators: 4 × 5 = 20
Step 2: Multiply denominators: 5 × 8 = 40
Step 3: Simplify 20⁄40: HCF is 20, so 20⁄40 = 1⁄2
Answer: 1⁄2
Calculate 11⁄2 ÷ 3⁄4
Step 1: Convert 1½ to an improper fraction: 3⁄2
Step 2: Keep, Change, Flip: 3⁄2 × 4⁄3
Step 3: Multiply: 12⁄6 = 2
Answer: 2
Knowledge Check
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Key Concepts Summary
- ●To add or subtract fractions, find the LCD, convert both fractions, then operate on the numerators.
- ●To multiply fractions, multiply numerators together and denominators together, then simplify.
- ●To divide fractions, use Keep, Change, Flip — multiply by the reciprocal of the second fraction.
- ●Always convert mixed numbers to improper fractions before multiplying or dividing.
- ●Always simplify your final answer by dividing the numerator and denominator by their HCF.