Can someone elaborate the rule of common denominators?

Question

Autumn Arnold · Accepted Answer

Only things that are the same can be added or subtracted. Hence fractions can only be added or subtracted with a common denominator.

Apples and Oranges can not be added. Change both to fruit ( a common denominator) Now Apples and Oranges can be added find to a total number of pieces of fruit. Fractions with a different denominator are like Apples and Oranges they can not be added. Change both fractions to fractions with the same denominator. Now the fractions can be aded or subtracted.

Julian Avila · Answer

Refer to the explanation.

When adding or subtracting fractions, they must have the same denominator; a common denominator. So what do we do if they don't have a common denominator? We can convert them into equivalent fractions by multiplying by a fractional form of #1;# a fraction with the same numerator and denominator. For example, #3/3=1#. This will change the numbers, but not the value of the fraction. Example: What is #3/8 + 1/4#? We can see that the denominators are not common. We can also see that the denominator #4# can become #8# by multiplying #1/4# by #2/2#. #3/8+1/4xxcolor(teal)(2/2# Simplify. #3/8+2/8# Now add the fractions. #3/8+2/8=5/8# Example: What is #3/12-2/15#? In this example, the denominators are not multiples of each other, so we must first determine the least common denominator (LCD) of #12# and #15#. List the multiples of #12# and #15#. The lowest number they have in common is the LCD. #12:##12,24,36,48,color(red)60,72,84...# #15:##15,30,45,color(red)60...# The LCD is #60#. Now go back to the question. #3/12-2/15# Now we must multiply each fraction by a fractional form of #1# that gives them a denominator of #60#. #12xx5=60# #15xx4=60# Multiply each fraction by the appropriate fractional form of #1#. #3/12xxcolor(teal)(5/5)-2/15xxcolor(magenta)(4/4# Simplify. #15/60-8/60# Simplify. #7/60# Since #7# is a prime number, #7/60# cannot be reduced.

Benjamin Altschul · Answer

undefined The rule of common denominators states that when adding or subtracting fractions, you need to have a common denominator. This common denominator is the same for all fractions involved in the operation. To achieve this:

1. Identify the least common multiple (LCM) of the denominators of the fractions.

2. Rewrite each fraction with the LCM as the common denominator by multiplying both the numerator and denominator of each fraction by the appropriate factor to make the denominators equal.

3. Once all fractions have the same denominator, you can add or subtract the numerators while keeping the common denominator unchanged.

4. Simplify the resulting fraction if necessary by reducing it to lowest terms.

By following the rule of common denominators, you ensure that fractions can be added or subtracted correctly without changing their values.