There are two cases regarding the denominators when we subtract ordinary fractions:
- A. the fractions have like denominators;
- B. the fractions have unlike denominators.
A. How to subtract ordinary fractions that have like denominators?
- Simply subtract 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 subtracting ordinary fractions that have like denominators, with explanations
3/18 + 4/18 - 5/18 = (3 + 4 - 5)/18 = 2/18;
- We simply subtracted the numerators of the fractions: 3 + 4 - 5 = 2;
- The denominator of the resulting fraction is: 18;
The resulting fraction is being reduced as: 2/18 = (2 ÷ 2)/(18 ÷ 2) = 1/9.
B. To subtract 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 (simplify them).
- Factor the numerator and the denominator of each fraction, break them down to prime factors (run their prime factorization).
- Calculate GCF, the greatest common factor of the numerator and of the denominator of each fraction.
- GCF is the product of all the unique common prime factors of the numerator and of the denominator, multiplied by the lowest exponents.
- Divide the numerator and the denominator of each fraction by their 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, also called the lowest common denominator (the least common denominator).
- 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, multiplied 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 the denominator of each fraction, in order to calculate each fraction's expanding number.
4. Expand each fraction:
- Multiply each fraction's both numerator and denominator by the expanding number.
- At this point, fractions are built up to the same denominator.
5. Subtract the fractions:
- In order to subtract all the fractions simply subtract 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.