Learn how to subtract ordinary fractions: theory, steps and a practical example. Fractions with different (like) or unlike denominators
How to: Subtracting ordinary (simple, common) math fractions. Steps.
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.
An example of subtracting fractions that have different denominators (unlike denominators), step by step explanations
6/90 + 16/24 - 30/75 = ?
1. Reduce the fractions to the lowest terms (simplify them):
6/90 = (2 × 3) / (2 × 32 × 5) = ((2 × 3) ÷ (2 × 3)) / ((2 × 32 × 5) ÷ (2 × 3)) = 1/(3 × 5) = 1/15
16/24 = 24 / (23 × 3) = (24 ÷ 23) / ((23 × 3) ÷ 23) = 2/3
30/75 = (2 × 3 × 5) / (3 × 52) = ((2 × 3 × 5) ÷ (3 × 5)) / ((3 × 25) ÷ (3 × 5)) = 2/5
The reduced fractions: 6/90 + 16/24 - 30/75 = 1/15 + 2/3 - 2/5
2. Calculate the least common multiple, LCM, of all the fractions' new denominators
- Factor all the denominators, break them down to their prime factors, then multiply all these prime factors, uniquely, by the largest exponents.
15 = 3 × 5
3 is already a prime number, it cannot be factored anymore
5 is a prime number, it cannot be factored anymore
LCM (15, 3, 5) = LCM (3 × 5, 3, 5) = 3 × 5 = 15
3. Calculate each fraction's expanding number:
4. Expand each fraction:
5. Subtract the fractions:
- Simply subtract the numerators of the fractions. The denominator = LCM.
6/90 + 16/24 - 30/75 = 1/15 + 2/3 - 2/5 = 1/15 + 10/15 - 6/15 = (1 + 10 - 6) / 15 = 5/15
6. Reduce the end fraction to the lowest terms, if needed.
-
5/15 = (5 ÷ 5)/(15 ÷ 5) = 1/3
More on ordinary (common) fractions / theory:
(7) Subtracting common (ordinary) fractions
Fractions operations that can be run automatically, with explanations:
Mathematical operations with fractions, plus the theory behind: addition, subtraction, multiplication, division, fractions reducing (simplifying), comparing, sorting in ascending order, converting numbers to fractions and percentages.