What is a Fraction?
A fraction represents a part of a whole: numerator ÷ denominator.
A fraction is written as n/d, where n is the numerator (how many parts you have) and d is the denominator (how many equal parts make the whole). The denominator tells you the “unit size”; the numerator tells you how many of those units you’re counting.
Fractions are a compact way to represent exact ratios. For example, 1/3 is exactly one third; writing it as a decimal (0.3333…) is only an approximation unless you use repeating notation.
- You ate 3 slices out of 8 → 3/8
- You finished 15 minutes of a 40-minute video → 15/40 = 3/8
Fast intuition: 1/2, 1/4, 3/4
A fraction is a ratio: numerator ÷ denominator.
How to Simplify Fractions
Simplify by dividing numerator and denominator by their greatest common divisor (GCD).
A fraction is in simplest form when the numerator and denominator share no common factor greater than 1. Simplifying does not change the value — it just changes how you write it.
The reliable method is: find the GCD of numerator and denominator, then divide both by that number. If you don’t know the GCD quickly, you can simplify step-by-step by canceling common factors (2, 3, 5, etc.).
- 24/36 → divide both by 12
- 24 ÷ 12 = 2
- 36 ÷ 12 = 3
- Result: 2/3
Simplifying is about shared factors
Simplify with GCD: (n/d) → (n÷g)/(d÷g).
Add & Subtract Fractions
Use a common denominator, then add/subtract the numerators.
To add or subtract fractions, you need the same denominator. The easiest safe approach is to use the common denominator d₁×d₂, convert both fractions, then add/subtract numerators.
After the operation, simplify the result. Many mistakes happen when people add denominators directly — don’t do that.
- 1/6 + 1/4
- Common denominator = 24
- 1/6 = 4/24, 1/4 = 6/24
- Sum = 10/24 = 5/12
Never add denominators
Add/subtract: make denominators equal, then combine numerators.
Multiply & Divide Fractions
Multiply: multiply across. Divide: multiply by the reciprocal.
Multiplying fractions is straightforward: multiply numerators together and denominators together. Then simplify.
Dividing by a fraction means multiplying by its reciprocal: a/b ÷ c/d = a/b × d/c. If the numerator of the divisor is 0, division is undefined.
- 2/3 × 9/10
- Multiply across: (2×9)/(3×10) = 18/30
- Simplify: 18/30 = 3/5
Cancel before multiplying (to keep numbers small)
Divide by a fraction by multiplying its reciprocal.
Compare Fractions (>, <, =)
Compare by cross-multiplying instead of converting to decimals.
To compare a/b and c/d, compare a×d with c×b. This avoids decimal rounding and stays exact.
If a×d is bigger than c×b, then a/b is bigger than c/d. If they’re equal, the fractions are equal.
- Which is bigger: 5/8 or 2/3?
- Cross-multiply: 5×3 = 15, 2×8 = 16
- 15 < 16 → 5/8 < 2/3
Decimals can mislead for repeating fractions
Compare a/b and c/d using a×d vs c×b.
Convert: Fraction ↔ Decimal ↔ Percent
Decimals and percents are just different representations of the same ratio.
To convert a fraction to a decimal, divide numerator by denominator. To convert to percent, multiply the decimal by 100 (or multiply the fraction by 100).
Not every fraction terminates as a decimal. Fractions like 1/2, 3/8, 7/20 terminate. Fractions like 1/3, 2/7 repeat.
- 3/8 = 0.375
- Percent: 0.375 × 100 = 37.5%
Quick test: when does a decimal terminate?
Decimal = numerator ÷ denominator; Percent = decimal × 100.
Mixed Numbers (and improper fractions)
A mixed number like 2 1/3 is the sum of a whole number and a fraction.
Mixed numbers are useful when the fraction is larger than 1. For example, 7/3 is commonly written as 2 1/3.
Convert mixed to improper by: whole×denominator + numerator, over the same denominator. Keep the sign consistent (a negative mixed number is treated as a negative whole part).
- 2 1/3 → (2×3 + 1)/3 = 7/3
- 7/3 → 2 remainder 1 → 2 1/3
Be careful with negative mixed numbers
Mixed → improper: (whole×d + n)/d.
Common Mistakes
Most errors come from mixing up denominators or skipping simplification.
The denominator is the “unit size”. When you add or subtract, you must convert both fractions to the same unit size. When you multiply/divide, keep track of reciprocal and simplify to avoid overflow.
A good habit is to simplify early and often. Also, if the denominator is 0, the fraction is undefined — that should be treated as an error.
- Wrong: 1/4 + 1/4 = 2/8
- Right: 1/4 + 1/4 = 2/4 = 1/2
Sanity check: does your answer make sense?
Use common denominators for +/−, reciprocals for ÷, and simplify results.
Quick Reference
Keep these patterns handy for most fraction tasks.
Simplify: divide numerator and denominator by GCD.
Add/subtract: common denominator → add/subtract numerators.
Multiply: multiply across, then simplify (cancel first if possible).
Divide: multiply by reciprocal (and check divisor isn’t 0).
Compare: cross-multiply for an exact comparison.
- a/b + c/d = (ad + bc) / bd
- a/b × c/d = ac / bd
- a/b ÷ c/d = ad / bc
| Operation | Formula | Notes |
|---|---|---|
| Add | a/b + c/d = (ad + bc) / bd | Simplify after |
| Subtract | a/b − c/d = (ad − bc) / bd | Simplify after |
| Multiply | a/b × c/d = ac / bd | Cancel before multiplying if possible |
| Divide | a/b ÷ c/d = ad / bc | c must not be 0 |
Most fraction work reduces to a small set of reliable formulas.
FAQs
What is the simplest way to simplify a fraction?
How do I add fractions with different denominators?
Why does 1/3 not terminate as a decimal?
How do I convert a fraction to percent?
Want to go deeper?
Open the full Fraction Calculator to explore step-by-step methods, recurring decimals, best approximations, and equivalent fractions.