There are two cases regarding the denominators when we add ordinary fractions:

• A. the fractions have like denominators;
• B. the fractions have unlike denominators.

A. How to add ordinary fractions that have like denominators?

• Simply add the numerators of the fractions.
• The denominator of the resulting fraction will be the common denominator of the fractions.
• Reduce the resulting fraction.

An example of adding ordinary fractions that have like denominators, with explanations

• ³/₁₈ + ⁴/₁₈ + ⁵/₁₈ = ^{(3 + 4 + 5)}/₁₈ = ¹²/₁₈;

• We simply added the numerators of the fractions: 3 + 4 + 5 = 12;
• The denominator of the resulting fraction is: 18;

• The resulting fraction is being reduced: ¹²/₁₈ = ^{(12 ÷ 6)}/_{(18 ÷ 6)} = ²/₃.

• How to reduce (simplify) the common (ordinary) fraction ¹²/₁₈?

B. To add fractions with different denominators (unlike denominators), build up the fractions to the same denominator. How is it done?

• 1. Reduce the fractions to the lowest terms (simplifying).

  • Factor the numerator and the denominator of each fraction down to prime factors (prime factorization).

  • Factor numbers online down to their prime factors.

  • Calculate GCF, the greatest common factor (also called GCD, greatest common divisor, HCF, greatest common factor) of each fraction's numerator and denominator.
  • GCF is the product of all the unique common prime factors of the numerator and the denominator, taken by the lowest exponents.

  • Calculate online the greatest common factor, GCF.

  • Divide the numerator and the denominator of each fraction by their greatest common factor, GCF - after this operation the fraction is reduced to its lowest terms equivalent.

  • Reduce (simplify) fractions online to their lowest terms, with explanations.

• 2. Calculate the least common multiple, LCM, of all the fractions' new denominators:

  • LCM is going to be the common denominator of the added fractions.
  • Factor all the new denominators of the reduced fractions (run the prime factorization).
  • The least common multiple, LCM, is the product of all the unique prime factors of the denominators, taken by the largest exponents.

  • Calculate LCM, the least common multiple of numbers.

• 3. Calculate each fraction's expanding number:

  • The expanding number is the non-zero number that will be used to multiply both the numerator and the denominator of each fraction, in order to build all the fractions up to the same common denominator.
  • Divide the least common multiple, LCM, calculated above, by each fraction's denominator, in order to calculate each fraction's expanding number.

• 4. Expand each fraction:

  • Multiply each fraction's both numerator and denominator by expanding number.
  • At this point, fractions are built up to the same denominator.

• 5. Add the fractions:

  • In order to add all the fractions simply add all the fractions' numerators.
  • The end fraction will have as a denominator the least common multiple, LCM, calculated above.

• 6. Reduce the end fraction to the lowest terms, if needed.

  • Reduce (simplify) ordinary fractions to lowest terms.

Read the rest of this article, here > How to add ordinary (common) fractions

(1) What is a fraction? Fractions types. How do they compare?

(2) Changing the form of fractions, by expanding or reducing (simplifying)

(3) How to reduce fractions (simplifying). The greatest common factor, GCF

(4) How to compare two fractions with unlike (different) numerators and denominators

(5) How to sort out fractions in ascending order

(6) Adding common (ordinary) fractions

(7) Subtracting common (ordinary) fractions

(8) Multiplying common (ordinary) fractions

(9) Fractions as rational numbers