# Fractions, theory: subtracting ordinary math fractions - how to subtract fractions with unlike (different) denominators?

## Tutoring and a practical example: subtracting multiple ordinary (simple, common) fractions

### To subtract fractions with different denominators (unlike denominators), fractions should be first brought to the same denominator. How is it done?

• #### 2. To bring all the fractions to the same denominator, you must calculate the fractions' denominators lowest common multiple, LCM:

• Fractions' denominators must be factored down to their constituent prime factors (denominators prime factorization).
• The denominators' lowest common multiple, LCM, will contain all of the unique prime factors, by the highest powers.
• Whether you don't know how or you'd like to practice the process, go to the address on the numere-prime.ro website: LCM, lowest common multiple of two numbers.
• #### 3. Calculate each fraction's expanding number, the non-zero number that will be used to multiply fraction's both numerator and denominator in order to bring all the fractions to the same common denominator:

• Divide the lowest common multiple, LCM, calculated above, by each fraction's denominator, getting the expanding number; that non-zero number will be used to multiply both the numerator and denominator of each fraction.
• #### 4. Expand fractions - multiply each fraction's both numerator and denominator by its expanding non-zero number.

• At this point, fractions are brought to the same denominator. In order to subtract the fractions simply subtract fractions' numerators. The end fraction will have as a denominator the lowest common multiple calculated above.

### An example of subtracting three ordinary fractions with different denominators (unlike denominators): 6/90 - 16/24 - 30/75

#### Reduce each fraction to lowest terms (simplify):

• Factor each fraction's both numerator and denominator down to their constituent prime factors (numerator and denominator prime factorization), then divide each fraction's both numerator and denominator by their greatest common factor, GCF (also called the greatest common divisor, GCD, or the highest common factor, HCF); GCF has only their unique common prime factors, by the lowest powers.
• Whether you don't know how to calculate the greatest common divisor of two numbers or you just need some refreshment, access this page on numere-prime.ro website: greatest common factor or divisor (GCF, GCD).
• Reduce the 1st fraction to lowest terms (simplify it):
• 6/90 = (2 * 3) / (2 * 32 * 5) = ((2 * 3) ÷ (2 * 3)) / ((2 * 32 * 5) ÷ (2 * 3)) = 1 / (3 * 5) = 1/15
• Reduce the 2nd fraction to lowest terms (simplify it):
• 16/24 = 24 / (23 * 3) = (24 ÷ 23) / ((23 * 3) ÷ 23) = 2/3
• Reduce the 3rd fraction to lowest terms (simplify it):
• 30/75 = (2 * 3 * 5) / (3 * 52) = ((2 * 3 * 5) ÷ (3 * 5)) / ((3 * 25) ÷ (3 * 5)) = 2/5
• At this point all the fractions are reduced (simplified): 6/90 - 16/24 - 30/75 = 1/15 - 2/3 - 2/5

#### Calculate the lowest common multiple, LCM

• Next, we calculate the lowest common multiple LCM of all three fractions' denominators. For that, factor each fraction's denominator down to its constituent prime factors (each fraction's denominator prime factorization); then take ALL the unique denominators' prime factors, by the highest powers.
• First fraction's denominator prime factorization: 15 = 3 * 5
• Second fraction's denominator prime factorization: 3 is already a prime number, it cannot be prime factorized
• Third fraction's denominator prime factorization: 5 is a prime number, it cannot be prime factorized
• The lowest common multiple LCM of all fractions' denominators must contain all of the unique denominators' prime factors by the highest powers:
• LCM (15; 3; 5) = LCM (3 * 5; 3; 5) = 3 * 5 = 15

#### Calculate each fraction's expanding number:

• Calculate each fraction's expanding number - this non-zero number will be used to multiply each fraction's both numerator and denominator. This number is calculated by dividing the lowest common multiple, LCM, by each fraction's denominator:
• first fraction expanding number: 15 ÷ 15 = 1
• second fraction expanding fraction: 15 ÷ 3 = 5
• third fraction expanding fraction: 15 ÷ 5 = 3

#### Bring the fractions to the same denominator:

• To bring the fractions to the same denominator, expand them (multiply each fraction's both numerator and denominator by its corresponding expanding number calculated above):
• the first fraction stays unchanged: 1/15 = (1 * 1)/(1 * 15) = 1/15
• the second fraction becomes: 2/3 = (5 * 2)/(5 * 3) = 10/15
• the third fraction becomes: 2/5 = (3 * 2)/(3 * 5) = 6/15

#### The final result of subtracting the fractions:

• 6/90 - 16/24 - 30/75 = 1/15 - 2/3 - 2/5 = 1/15 - 10/15 - 6/15 = - 15/15 = - 1
• The end fraction is an improper one - the absolute value of the numerator is equal to the absolute value of the denominator.