A fraction expresses a division: 3/4 means "three parts out of four", that is 0.75. The four operations with fractions follow precise and different rules: addition and subtraction require the same denominator, while multiplication and division do not. The calculator above performs every operation and simplifies the result; below you'll find the method for each, with checkable examples.
Addition and subtraction: you need the same denominator
To add or subtract two fractions you must first give them the same denominator. With 1/2 + 1/3: the least common denominator between 2 and 3 is 6, so 1/2 = 3/6 and 1/3 = 2/6; the sum is 3/6 + 2/6 = 5/6. In the same way 3/4 − 1/6 = 9/12 − 2/12 = 7/12.
Multiplication: numerator times numerator
Multiplication is the simplest: you multiply the numerators together and the denominators together. 2/3 × 4/5 = (2×4)/(3×5) = 8/15. No common denominator needed.
Division: multiply by the inverted fraction
Dividing by a fraction is the same as multiplying by its reciprocal (the flipped fraction). 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8, that is 1 and 7/8. The rule is summed up as "flip the second one and multiply".
Simplifying the result
A fraction is simplified by dividing numerator and denominator by their Greatest Common Divisor. 8/12 has a GCD of 4, so it becomes 2/3. A fraction is in "lowest terms" when numerator and denominator share no common divisor other than 1.
From fraction to decimal (and back)
To get the decimal value, just divide the numerator by the denominator: 5/8 = 5 ÷ 8 = 0.625. Conversely, 0.75 is written 75/100, which simplified is 3/4.
Equivalent fractions and comparison
Two fractions are equivalent when they represent the same value: 1/2, 2/4 and 50/100 are the same quantity. You get them by multiplying or dividing numerator and denominator by the same number. To compare two fractions it helps to give them the same denominator: between 3/5 and 5/8, with denominator 40 they become 24/40 and 25/40, so 5/8 is the larger.
Improper fractions and mixed numbers
A fraction is improper when the numerator is larger than the denominator, for example 7/4. It can be written as a mixed number by dividing: 7 ÷ 4 is 1 with remainder 3, so 7/4 = 1 and 3/4. The reverse step: 2 and 1/3 = (2×3 + 1)/3 = 7/3.
Fractions in real life
- Recipes: halving a 3/4-cup amount means computing 3/4 × 1/2 = 3/8 of a cup.
- Measurements: 1/2 metre plus 1/4 metre is 2/4 + 1/4 = 3/4 of a metre.
- Time: a quarter of an hour plus half an hour is 1/4 + 2/4 = 3/4 of an hour, that is 45 minutes.
Common mistakes to avoid
The most common mistake is adding numerators and denominators separately: 1/2 + 1/3 does not equal 2/5. You must always reduce to the same denominator first. Another mistake is forgetting to simplify the final result, leaving for example 8/12 instead of 2/3.
Frequently asked questions
What does least common denominator mean?
It's the smallest number divisible by both denominators. For 1/4 and 1/6 it's 12, because 12 is the smallest common multiple of 4 and 6.
How do you add fractions and whole numbers?
You turn the whole number into a fraction with denominator 1: 2 + 1/3 = 6/3 + 1/3 = 7/3, that is 2 and 1/3.
Can a fraction have a denominator of zero?
No: the denominator says how many parts you divide into, and dividing by zero is undefined. A fraction with denominator 0 has no mathematical meaning.