Free Tool · Auto-Simplify · Step-by-Step

Fraction Calculator

Add, subtract, multiply and divide fractions. Auto-simplified, with mixed-number and decimal forms.

Operation
First fraction
/
Second fraction
/

What this fraction calculator does

This tool performs the four basic arithmetic operations on fractions and returns the answer in its simplest form. You enter two fractions — numerator and denominator for each — choose whether to add, subtract, multiply, or divide, and the calculator produces the exact result, the simplified fraction, the decimal value, and (where meaningful) the mixed-number form. It shows the full step-by-step working, so you can see the common denominator being found, the numerators being combined, and the final simplification by the greatest common divisor. Negative fractions are supported, and the calculator normalises the sign so the denominator is always positive in the result.

Formulas used in each operation

For two fractions a/b and c/d, the four operations are defined as follows.

Addition
a/b + c/d = (a·d + c·b) / (b·d)
Subtraction
a/b − c/d = (a·d − c·b) / (b·d)
Multiplication
a/b × c/d = (a·c) / (b·d)
Division
a/b ÷ c/d = a/b × d/c = (a·d) / (b·c)
Simplification
Let g = gcd(|numerator|, denominator). Then n/d simplifies to (n/g) / (d/g).

Variable definitions

  • a — numerator of the first fraction.
  • b — denominator of the first fraction (must be non-zero).
  • c — numerator of the second fraction.
  • d — denominator of the second fraction (must be non-zero).
  • gcd(n, d) — greatest common divisor of numerator and denominator.

Worked examples

Example 1: 1/2 + 1/3

  1. Common denominator: 2 × 3 = 6.
  2. Convert: 1/2 = 3/6 and 1/3 = 2/6.
  3. Add numerators: 3 + 2 = 5.
  4. Result: 5/6. Decimal value: 0.833333…

Example 2: 3/4 − 1/6

  1. Least common denominator is 12.
  2. Convert: 3/4 = 9/12 and 1/6 = 2/12.
  3. Subtract: 9 − 2 = 7.
  4. Result: 7/12. Decimal value: 0.583333…

Example 3: 2/3 × 9/4

  1. Multiply numerators: 2 × 9 = 18.
  2. Multiply denominators: 3 × 4 = 12.
  3. Result before simplification: 18/12.
  4. Divide both by gcd(18, 12) = 6: 3/2.
  5. Mixed form: 1 1/2. Decimal value: 1.5

Example 4: 3/5 ÷ 2/7

  1. Flip the divisor: 2/7 → 7/2.
  2. Multiply: 3/5 × 7/2 = 21/10.
  3. gcd(21, 10) = 1, so the fraction is already in lowest terms.
  4. Mixed form: 2 1/10. Decimal value: 2.1

Example 5: 18/24 simplified

  1. Find gcd(18, 24). Euclid's algorithm: gcd(24, 18) → gcd(18, 6) → gcd(6, 0) = 6.
  2. Divide: 18/6 = 3, 24/6 = 4.
  3. Result: 3/4, which is the same fraction in lowest terms.

Understanding fractions in Class 9 & 10

Fractions are the foundation of several topics in the Punjab board syllabus. In Class 9 algebra, you manipulate algebraic fractions with variables in numerator and denominator. In Class 10, fractions appear in the quadratic formula, in probability, and in trigonometry via ratios. Being able to add and simplify fractions quickly and correctly prevents most errors in later chapters. Two key ideas to remember:

  • Common denominators before adding or subtracting. You cannot directly add halves to thirds. Multiplying the two denominators always gives a common one, even if it is not the smallest.
  • Multiply straight across for multiplication and division. No common denominator is needed. For division, flip the second fraction and multiply.

Common mistakes to avoid

  • Adding numerators and denominators directly. 1/2 + 1/3 is not 2/5. You must find a common denominator first.
  • Forgetting to simplify. An answer like 18/12 is not considered "finished" in exams. Always reduce to lowest terms and, if it's an improper fraction, consider the mixed form.
  • Sign errors when subtracting. Subtracting a negative fraction becomes addition: 1/2 − (−1/3) = 1/2 + 1/3 = 5/6.
  • Flipping the wrong fraction when dividing. You flip the divisor (the second fraction), not the dividend. 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8.
  • Zero denominators. Any fraction with denominator 0 is undefined. 5/0 has no value. Even 0/0 is undefined, not zero.
  • Mixing up improper fractions and mixed numbers. 7/4 and 1 3/4 are the same value. Learn to convert both ways.

FAQ

How do I add fractions with different denominators?

Find a common denominator, convert both fractions, then add the numerators. For 1/2 + 1/3, the least common denominator is 6, giving 3/6 + 2/6 = 5/6. This calculator does that automatically and shows each step.

Why does dividing fractions involve flipping the second one?

Dividing by a fraction is the same as multiplying by its reciprocal. So a/b ÷ c/d equals a/b × d/c, which is ad/bc. The flip is not a magic rule — it follows from the definition of division.

What is the difference between an improper fraction and a mixed number?

An improper fraction has a numerator greater than or equal to its denominator, like 7/4. A mixed number writes the same value as a whole number plus a proper fraction, like 1 3/4. They are the same quantity in different notation.

How does the calculator simplify a fraction?

It divides the numerator and denominator by their greatest common divisor using the Euclidean algorithm. For 18/24, the GCD is 6, giving 3/4. A fraction is in lowest terms when the GCD of numerator and denominator is 1.

Can I enter negative fractions?

Yes. Put the minus sign in front of either the numerator or the whole fraction. The calculator normalizes the sign so the denominator stays positive. For example -3/4 and 3/-4 both become -3/4 in the result.

What happens if I enter a zero denominator?

A fraction with denominator zero is undefined, so the calculator will show an error. Zero numerators are fine — 0/5 equals 0. Division by a fraction whose numerator is zero, like 3/4 ÷ 0/5, is also an error because it is equivalent to dividing by zero.

Related tools

Related Hira Academy resources

Fractions are used in nearly every chapter of the BISE Punjab mathematics syllabus. Practise the basics with our free Class 9 notes and Class 10 notes, especially the chapters on rational expressions and algebraic fractions.

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