Fraction Calculator: Complete Guide to Fraction Arithmetic, Simplification & Conversion
What Is a Fraction?
A fraction is a mathematical expression representing part of a whole. It takes the form n/d where n is the numerator (how many parts you have) and d is the denominator (how many equal parts the whole is divided into). For example, 3/8 means 3 out of 8 equal parts.
Fractions are classified as:
- Proper fraction: numerator < denominator (e.g. 3/4). Value is between 0 and 1.
- Improper fraction: numerator ≥ denominator (e.g. 7/4). Value is ≥ 1.
- Mixed number: a whole number plus a proper fraction (e.g. 1¾). Equivalent to an improper fraction.
- Unit fraction: numerator = 1 (e.g. 1/5). The building block of all fractions.
Fraction Arithmetic
The four arithmetic operations on fractions follow specific rules:
Addition and Subtraction
a/b ± c/d = (a×d ± c×b) / (b×d)
Find the LCD (LCM of denominators), convert both fractions, then add or subtract numerators. For example: 1/3 + 1/4 → LCD = 12 → 4/12 + 3/12 = 7/12.
Multiplication
a/b × c/d = (a×c) / (b×d)
Simply multiply the numerators together and the denominators together. Tip: simplify (cross-cancel) before multiplying to keep numbers small. For example: 2/3 × 3/4 → cancel 3s → 2/4 = 1/2.
Division (Keep-Change-Flip)
a/b ÷ c/d = a/b × d/c = (a×d) / (b×c)
Multiply the first fraction by the reciprocal of the second. For example: 3/5 ÷ 2/7 = 3/5 × 7/2 = 21/10 = 2 1/10.
Simplifying Fractions Using GCD
A fraction is in simplest form (lowest terms) when GCD(numerator, denominator) = 1. To simplify, divide both by their Greatest Common Divisor.
simplified = (n ÷ GCD) / (d ÷ GCD)
The Euclidean Algorithm finds GCD efficiently: divide the larger number by the smaller, take the remainder, and repeat. When the remainder is 0, the last non-zero remainder is the GCD. For example, GCD(48, 18): 48 mod 18 = 12; 18 mod 12 = 6; 12 mod 6 = 0 → GCD = 6. Therefore 48/18 = 8/3.
Fraction Conversion
A fraction can be expressed in several equivalent forms:
- Decimal: divide numerator by denominator — 3/8 = 0.375
- Percent: multiply decimal by 100 — 3/8 = 37.5%
- Mixed number: floor(n/d) whole + remainder/d — 7/4 = 1¾
- Reciprocal: flip numerator and denominator — reciprocal of 3/4 is 4/3
Terminating vs repeating decimals: A fraction produces a terminating decimal if and only if its simplified denominator has no prime factors other than 2 and 5. Otherwise, the decimal repeats indefinitely (e.g. 1/3 = 0.333…, 1/7 = 0.142857142857…).
Mixed Number Arithmetic
To operate on mixed numbers, convert them to improper fractions first:
w n/d → (w × d + n) / d
Perform the operation, then convert back: 1½ + 2⅓ = 3/2 + 7/3 = 9/6 + 14/6 = 23/6 = 3⅚.
A mixed number is always interpreted as addition of the whole number and the fraction — 2¾ means 2 + 3/4, never 2 × 3/4.
Comparing Fractions & Least Common Denominator
Three methods to compare fractions a/b and c/d:
- Cross-multiplication: if a×d > c×b, then a/b > c/d. Fast, no fractions needed. Example: 2/3 vs 3/4 → 2×4=8 vs 3×3=9 → 2/3 < 3/4.
- LCD method: LCD = LCM(b, d). Convert both fractions to the LCD and compare numerators. 2/3 and 3/4 with LCD=12: 8/12 vs 9/12 → 2/3 < 3/4.
- Decimal comparison: a/b = a÷b; c/d = c÷d. Compare the decimals directly.
The Least Common Multiple (LCM) of two numbers a and b is: LCM(a,b) = |a × b| / GCD(a, b). For example: LCM(4, 6) = 24/GCD(4,6) = 24/2 = 12.
Frequently Asked Questions
What is a fraction?
A fraction represents a part of a whole. It consists of a numerator (top number) and a denominator (bottom number). The denominator tells you how many equal parts the whole is divided into, while the numerator tells you how many of those parts you have. For example, 3/4 means 3 parts out of 4 equal parts.
How do I add fractions with different denominators?
To add fractions with different denominators, find the Least Common Denominator (LCD), which is the Least Common Multiple (LCM) of the two denominators. Convert each fraction to an equivalent fraction with the LCD as the new denominator, then add the numerators. For example: 1/2 + 1/3 → LCD = 6 → 3/6 + 2/6 = 5/6.
How do I divide fractions?
To divide fractions, use the "Keep-Change-Flip" (KCF) rule: keep the first fraction unchanged, change the division sign to multiplication, and flip (take the reciprocal of) the second fraction. For example: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6. Then simplify if possible.
What is a mixed number?
A mixed number combines a whole number and a proper fraction, such as 2¾ (two and three-quarters). It equals 2 + 3/4 = 11/4 as an improper fraction. To convert: multiply the whole number by the denominator and add the numerator — for 2¾: (2×4 + 3)/4 = 11/4.
How do I simplify a fraction to its lowest terms?
To simplify a fraction, find the Greatest Common Divisor (GCD) of the numerator and denominator, then divide both by it. For example: 12/18 → GCD(12, 18) = 6 → (12÷6)/(18÷6) = 2/3. Use the Euclidean algorithm for efficiency: repeatedly divide the larger by the smaller and take the remainder until the remainder is 0.
What is the Least Common Denominator (LCD)?
The Least Common Denominator (LCD) is the smallest number that is a multiple of all the denominators in a set of fractions. It equals the Least Common Multiple (LCM) of the denominators. For 1/4 and 1/6, LCD = LCM(4, 6) = 12. The LCD is used to add or subtract fractions with different denominators.
How do I convert a fraction to a decimal?
To convert a fraction to a decimal, divide the numerator by the denominator. For example: 3/4 = 3 ÷ 4 = 0.75. The decimal may terminate (1/4 = 0.25) or repeat infinitely (1/3 = 0.3333…). Fractions whose simplified denominators have only 2 and 5 as prime factors always produce terminating decimals.
Can a fraction have a negative numerator or denominator?
Yes. A fraction can have negative values. A negative fraction has a negative sign, which by convention is placed with the numerator: -3/4. If the denominator is negative, multiply both numerator and denominator by -1 to move the sign to the numerator: 3/(-4) = -3/4. The fraction -3/4, (-3)/4, and 3/(-4) are all equivalent.
How do I compare two fractions?
There are three methods: (1) Cross-multiplication: multiply each numerator by the opposite denominator and compare — if n1×d2 > n2×d1, then n1/d1 > n2/d2. (2) LCD method: convert both fractions to the same denominator then compare numerators. (3) Decimal method: divide each numerator by its denominator and compare the decimals.
What is an improper fraction?
An improper fraction has a numerator greater than or equal to its denominator, such as 7/4 or 5/5. Improper fractions are greater than or equal to 1. They can always be converted to a mixed number: 7/4 = 1¾. Both forms are mathematically equivalent and equally valid — the preferred form depends on context.