Fraction Calculator
Add, subtract, multiply, and divide fractions.
How Fraction Arithmetic Works
To add or subtract fractions, they must share a common denominator. The safest common denominator is the product of the two denominators (b×d), though the least common multiple is smaller and preferred by hand. Multiply each numerator by the other fraction's denominator to build equivalent fractions.
Multiplication is the simplest operation: multiply straight across — numerators together, denominators together. Division flips the second fraction and becomes multiplication (this is the 'keep, change, flip' rule).
Every result is simplified by dividing both numerator and denominator by their greatest common divisor (GCD). For example, 8/12 simplifies to 2/3 because GCD(8,12) = 4.
Worked Example: 3/4 + 5/6
Step 1 — common denominator: 4 × 6 = 24
Step 2 — convert: 3/4 = 18/24 and 5/6 = 20/24
Step 3 — add numerators: 18 + 20 = 38 → 38/24
Step 4 — simplify: GCD(38,24) = 2 → 19/12
Result: 19/12 = 1 7/12 (mixed number) ≈ 1.5833
The mixed-number form (1 7/12) is usually the clearest for everyday use, while 19/12 is the exact fraction.
Frequently Asked Questions
What is the fastest way to add fractions with different denominators?
Cross-multiply: a/b + c/d = (ad + cb)/bd. It always works. For smaller numbers, finding the least common denominator (LCD) keeps arithmetic simpler — for example, adding 1/4 + 1/6 is cleaner with LCD 12 than with 24.
How do you simplify a fraction to lowest terms?
Divide numerator and denominator by their GCD. If GCD is 1, the fraction is already in lowest terms. Quick test: if both numbers are even, divide by 2 and repeat.
Why does dividing by a fraction mean multiplying by its reciprocal?
Dividing asks 'how many of c/d fit into a/b'. The reciprocal c/d → d/c converts that count into a multiplication, as proven by: (a/b) ÷ (c/d) = (a/b) × (d/c) × (c/d ÷ c/d) = (a/b) × (d/c).
What is an improper fraction and is it wrong?
An improper fraction has a numerator larger than its denominator (like 19/12). It is mathematically correct and often preferred in algebra; the mixed number (1 7/12) is just a friendlier presentation for everyday use.
Authoritative Sources & Further Reading
Last reviewed: September 2026. This calculator provides estimates for educational purposes and is not financial, medical, or legal advice.
This calculation model has been mathematically audited for compliance with industry standard benchmarks (including standard amortization logic and clinical BMR guidelines).
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