First fraction
Operation
Second fraction
Result with steps (click to copy)
Working With Fractions
Three input formats accepted
Enter a plain fraction (3/4), a decimal (1.5), or a mixed number (1 1/2, with a space between the whole number and the fraction) — the calculator recognizes all three.
Results are always shown simplified
The output reduces to lowest terms automatically and shows the mixed-number form when the result is greater than 1, alongside the decimal equivalent.
Step-by-step working matches how you'd show it by hand
The common-denominator (for +/−) or numerator/denominator (for ×/÷) breakdown mirrors standard classroom method, useful for checking your own hand-worked steps.
How to Use the Fraction Calculator
- Enter first fraction — type the numerator (top) and denominator (bottom) separately.
- Select operation — choose addition, subtraction, multiplication, or division.
- Enter second fraction — or enter a whole number (the calculator treats it as a fraction with denominator 1).
- Click Calculate — see the result as a fraction, simplified fraction, mixed number, and decimal.
- Read the steps — the calculator shows the full working including finding common denominators for addition/subtraction.
➕ Fraction addition rule: You CANNOT add numerators unless denominators are the same. 1/3 + 1/4 ≠ 2/7. Find the LCD (12), convert: 4/12 + 3/12 = 7/12. The most common fraction error is adding numerators and denominators directly — always find a common denominator first for addition and subtraction.
Understanding Fraction Operations
➕ Addition/Subtraction
Find LCD (Lowest Common Denominator), convert both fractions to that denominator, add/subtract numerators, keep denominator, simplify. 1/3 + 1/4: LCD=12. 1/3=4/12, 1/4=3/12. 4/12+3/12=7/12. No further simplification needed (GCD of 7 and 12 is 1).
✖️ Multiplication
Multiply numerators together, multiply denominators together, simplify. 2/3 × 3/4 = (2×3)/(3×4) = 6/12 = 1/2. No common denominator needed! Cross-cancel before multiplying to simplify: 2/3 × 3/4 — cancel 3s: 2/1 × 1/4 = 2/4 = 1/2. Cross-cancelling makes multiplication much easier with large numbers.
➗ Division
'Keep, Change, Flip' — keep the first fraction, change division to multiplication, flip (take reciprocal of) the second fraction. 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8. Division by a fraction is equivalent to multiplication by its reciprocal — this rule works because both operations are inverses.
🔢 Simplification
Divide numerator and denominator by their GCD (Greatest Common Divisor). GCD(12,18)=6, so 12/18 = 2/3. Euclid's algorithm finds GCD efficiently: GCD(a,b) = GCD(b, a mod b), repeat until remainder=0. A fraction is fully simplified when GCD(numerator, denominator)=1.
📊 Mixed Numbers
A whole number plus a proper fraction: 2 3/4. Convert to improper fraction: (2×4+3)/4 = 11/4. Convert back: 11÷4 = 2 remainder 3 = 2 3/4. Always convert mixed numbers to improper fractions before performing operations — mixed number arithmetic is error-prone without this step.
🌐 Real Applications
Cooking: 2/3 cup × 3 servings. Construction: 3 1/4 inches + 2 3/8 inches. Music: 3/4 time signature = 3 beats per measure. Finance: fractional shares, interest rates. Probability: P(A)=3/8, P(B)=1/4, P(A and B)=3/32. Fractions are fundamental to everyday measurements and calculations.
Fractions in Mathematics and Daily Life
LCD vs GCD — two essential tools
LCD (Lowest Common Denominator) is needed for fraction addition and subtraction — it's the smallest number both denominators divide into. GCD (Greatest Common Divisor) is needed for simplifying fractions — it's the largest number that divides both numerator and denominator. These two concepts, along with the four fraction operations, cover essentially all elementary fraction arithmetic. Finding LCD: list multiples of each denominator until they match, or use LCD = (a×b)/GCD(a,b). Finding GCD: use Euclid's algorithm or list factors.
Fractions in algebra
Algebraic fractions (rational expressions) follow the same rules as numeric fractions. x/y + a/b = (xb+ay)/(yb). (x/y)÷(a/b) = x/y × b/a = xb/(ya). Simplify by factoring numerator and denominator and cancelling common factors. The main additional challenge: determining when fractions are undefined (denominator = 0) and finding restrictions on variables (x ≠ 2 if denominator contains x−2). Fraction arithmetic is the foundation for solving rational equations.
Decimal to fraction conversion
Terminating decimals: 0.75 = 75/100 = 3/4. 0.125 = 125/1000 = 1/8. Repeating decimals: 0.333... = 1/3. 0.142857142857... = 1/7. For repeating decimals, set x = 0.333..., then 10x = 3.333..., subtract: 9x = 3, x = 1/3. Irrational numbers (π, √2) cannot be expressed as fractions — they are infinite non-repeating decimals. Understanding this distinction is fundamental to number theory.
➗ Fraction operation memory aid: Add/Subtract: need common denominator (LCD method). Multiply: straight across (numerator × numerator, denominator × denominator). Divide: keep-change-flip (KCF). Simplify: divide by GCD. Mixed numbers: convert to improper fractions first. These five rules cover all fraction arithmetic. Mastering them makes algebra, probability, and advanced mathematics significantly more accessible.
Avoid These Mistakes
Adding denominators directly
1/2 + 1/3 is NOT 2/5 — you must find a common denominator first (1/2 + 1/3 = 3/6 + 2/6 = 5/6). Denominators are never simply added or multiplied together for addition/subtraction.
Forgetting to flip for division
Dividing by a fraction means multiplying by its reciprocal — a/b ÷ c/d = a/b × d/c. Skipping the flip is one of the most common fraction errors.
Leaving a mixed number unconverted before multiplying
Convert mixed numbers to improper fractions before multiplying or dividing — multiplying the whole-number and fraction parts separately gives a wrong result.
Frequently Asked Questions
How do I add fractions?
Find a common denominator, convert both fractions, then add the numerators. Example: 1/3 + 1/4. Common denominator = 12. 1/3 = 4/12, 1/4 = 3/12. Sum = 7/12. This calculator shows each step.
How do I multiply fractions?
Multiply numerators together and denominators together, then simplify. Example: 2/3 × 3/4 = 6/12 = 1/2. Much simpler than addition — no common denominator needed.
How do I divide fractions?
Multiply by the reciprocal of the second fraction. Flip the second fraction and multiply. Example: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6.
What is a mixed number?
A mixed number has a whole number part and a fraction part, like 2¾. This calculator automatically converts improper fractions (like 11/4) to mixed numbers (2¾) in the result — and you can also type a mixed number directly as input, like "2 3/4" (with a space between the whole number and the fraction).
References
📄 Your math textbook or course
Fraction arithmetic rules (common denominators, keep-change-flip for division) are standard across virtually every math curriculum — this calculator follows the same conventions taught in most classrooms.