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How to Simplify a Fraction: Step by Step

Hesaplayıcı3 min read

Short answer

To simplify a fraction, divide the numerator and the denominator by their greatest common divisor (GCD), the largest number that divides both. For 48/60 the GCD is 12: 48 ÷ 12 = 4 and 60 ÷ 12 = 5, so the result is 4/5. Simplifying does not change the value; both fractions equal 0.8.

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Formula

  • GCD = gcd(|numerator|, denominator)
  • simplified numerator = numerator / GCD
  • simplified denominator = denominator / GCD

What the terms mean:

  • numerator: the number above the fraction bar
  • denominator: the number below the fraction bar; it cannot be zero
  • GCD: the largest whole number that divides both without a remainder
  • |numerator|: the absolute value of the numerator; for a negative fraction the GCD is found without the sign

You can find the GCD with Euclid’s algorithm: divide the larger number by the smaller one, then divide the divisor by the remainder. When the remainder is 0, the last divisor is the GCD.

Step-by-step example

Simplify 48/60.

  1. Divide 60 by 48: the quotient is 1 and the remainder is 12.
  2. Divide 48 by 12: the quotient is 4 and the remainder is 0. So the GCD is 12.
  3. Divide the numerator by the GCD: 48 / 12 = 4.
  4. Divide the denominator by the GCD: 60 / 12 = 5.
  5. The simplified fraction is 4/5. Its decimal value is 0.8.

Turning an improper fraction into a mixed number

A fraction whose numerator is equal to or larger than its denominator is an improper fraction. Take 22/8:

  1. Simplify first: gcd(22, 8) = 2, so the fraction becomes 11/4.
  2. Divide the simplified numerator by the simplified denominator: 4 goes into 11 twice, with 3 left over. The whole part is 2 and the remainder is 3.
  3. Write the result: 22/8 = 11/4 = 2 3/4. Its decimal value is 2.75.

Example fractions

Fraction GCD Simplified Decimal
48/60 12 4/5 0.8
84/126 42 2/3 about 0.667
22/8 2 11/4, or 2 3/4 2.75
6/−8 2 −3/4 −0.75
7/9 1 7/9, already simplified about 0.778

Common mistakes

  • Stopping at a small common divisor. If you divide 48/60 by 2 and stop at 24/30, the fraction still simplifies, because gcd(24, 30) = 6. Dividing by the GCD finishes the job in one step.
  • Adding or subtracting the same number. Adding the same number to the numerator and the denominator changes the value. Multiplying or dividing both by the same number keeps it.
  • Leaving the minus sign in the denominator. Since a / −b = −a / b, the fraction 6/−8 is written −3/4.
  • Using zero as the denominator. Division by zero is undefined, so the denominator cannot be zero.

Worked example: Simplify 84/126

Decimal value

0.67

Simplified numerator
2
Simplified denominator
3
GCD (greatest common divisor)
42
Whole part
0
Remainder (numerator of the mixed number)
2
Step by step

  1. GCD (greatest common divisor)

    GCD = gcd(|numerator|, denominator)

    GCD = gcd(84, 126) = 42

  2. Simplified numerator

    simplified numerator = numerator / GCD

    simplified numerator = 84 / 42 = 2

  3. Simplified denominator

    simplified denominator = denominator / GCD

    simplified denominator = 126 / 42 = 3

  4. Decimal value

    decimal = numerator / denominator

    decimal = 84 / 126 = 0.67

  5. Whole part

    whole part = (simplified numerator − remainder) / simplified denominator

    whole part = (2 − 2) / 3 = 0

  6. Remainder (numerator of the mixed number)

    remainder = simplified numerator − whole part × simplified denominator

    remainder = 2 − 0 × 3 = 2

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