Mathsā± 5 min read
How to Add, Subtract, Multiply and Divide Fractions
Fractions trip up adults as often as children. Here's a clean, step-by-step guide to every operation ā including the shortcuts most people were never taught at school.
Fraction arithmetic has a reputation for being confusing, but it follows simple and consistent rules. Master these four operations and fractions become straightforward.
Key Vocabulary
Fraction: a/b
a = numerator (top number)
b = denominator (bottom number)
Proper fraction: numerator < denominator (e.g. 3/4)
Improper fraction: numerator ā„ denominator (e.g. 7/3)
Mixed number: whole number + fraction (e.g. 2 1/3)
Multiplying Fractions (Easiest Operation)
Multiply numerators together, multiply denominators together. That's it.
a/b Ć c/d = (aĆc) / (bĆd)
3/4 Ć 2/5 = (3Ć2) / (4Ć5) = 6/20 = 3/10
Always simplify by dividing numerator and denominator
by their greatest common factor (GCF).
GCF of 6 and 20 is 2 ā 6/20 = 3/10
Dividing Fractions (Flip and Multiply)
Dividing by a fraction is the same as multiplying by its reciprocal (flipped version).
a/b Ć· c/d = a/b Ć d/c = (aĆd) / (bĆc)
3/4 Ć· 2/5 = 3/4 Ć 5/2 = 15/8 = 1 7/8
Adding and Subtracting Fractions (Common Denominator)
To add or subtract fractions, they must have the same denominator. If they already do, just add/subtract the numerators.
Same denominator:
3/8 + 2/8 = 5/8
Different denominators ā find the LCM:
1/4 + 1/6
LCM of 4 and 6 = 12
Convert: 1/4 = 3/12 and 1/6 = 2/12
Add: 3/12 + 2/12 = 5/12
Finding the Lowest Common Multiple (LCM)
Method 1 (small numbers): list multiples until they match
Multiples of 4: 4, 8, 12, 16...
Multiples of 6: 6, 12, 18...
LCM = 12 ā
Method 2 (larger numbers): LCM = (a Ć b) Ć· GCF(a,b)
LCM(8, 12) = (8 Ć 12) Ć· GCF(8,12) = 96 Ć· 4 = 24
Working With Mixed Numbers
Convert to improper fractions first, operate, then convert back.
Convert mixed ā improper:
2 3/4 = (2Ć4 + 3)/4 = 11/4
Example: 2 3/4 + 1 1/3
= 11/4 + 4/3
LCM(4,3) = 12
= 33/12 + 16/12 = 49/12
= 4 1/12
Convert improper ā mixed:
49/12: 12 goes into 49 four times remainder 1 ā 4 1/12
Cross-Cancelling (A Useful Shortcut)
Before multiplying, you can cancel common factors between any numerator and any denominator ā even across different fractions. This keeps numbers smaller and avoids large simplification steps at the end.
4/9 Ć 3/8
Cross-cancel: 4 and 8 share factor 4 ā becomes 1 and 2
3 and 9 share factor 3 ā becomes 1 and 3
= 1/3 Ć 1/2 = 1/6
Versus the long way: (4Ć3)/(9Ć8) = 12/72 = 1/6 ā
Same answer, but cross-cancelling avoids large numbers.
Fractions, Decimals and Percentages
Fraction ā Decimal: divide numerator by denominator
3/4 = 3 Ć· 4 = 0.75
Decimal ā Fraction: use place value
0.75 = 75/100 = 3/4 (simplify by Ć·25)
Percentage ā Fraction: put over 100, simplify
35% = 35/100 = 7/20