How to: Adding ordinary (simple, common) math fractions. Steps.
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.
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).
- 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.
- 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.
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.
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.