4.5 Add and subtract fractions with different denominators  (Page 2/7)

Find the LCD for the fractions $\frac{7}{12}$ and $\frac{5}{18}.$

Solution

Factor each denominator into its primes.
List the primes of 12 and the primes of 18 lining them up in columns when possible.
Bring down the columns.
Multiply the factors. The product is the LCM.	$\text{LCM}=36$
The LCM of 12 and 18 is 36, so the LCD of $\frac{7}{12}$ and $\frac{5}{18}$ is 36.	LCD of $\frac{7}{12}$ and $\frac{5}{18}$ is 36.

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To find the LCD of two fractions, find the LCM of their denominators. Notice how the steps shown below are similar to the steps we took to find the LCM.

Convert fractions to equivalent fractions with the lcd

Earlier, we used fraction tiles to see that the LCD of $\frac{1}{4}\text{and}\frac{1}{6}$ is $12.$ We saw that three $\frac{1}{12}$ pieces exactly covered $\frac{1}{4}$ and two $\frac{1}{12}$ pieces exactly covered $\frac{1}{6},$ so

We say that $\frac{1}{4}\text{and}\frac{3}{12}$ are equivalent fractions and also that $\frac{1}{6}\text{and}\frac{2}{12}$ are equivalent fractions.

We can use the Equivalent Fractions Property to algebraically change a fraction to an equivalent one. Remember, two fractions are equivalent if they have the same value. The Equivalent Fractions Property is repeated below for reference.

To add or subtract fractions with different denominators, we will first have to convert each fraction to an equivalent fraction with the LCD. Let’s see how to change $\frac{1}{4}\text{and}\frac{1}{6}$ to equivalent fractions with denominator $12$ without using models.

Convert $\frac{1}{4}\text{and}\frac{1}{6}$ to equivalent fractions with denominator $12,$ their LCD.

Solution

Find the LCD.	The LCD of $\frac{1}{4}$ and $\frac{1}{6}$ is 12.
Find the number to multiply 4 to get 12.
Find the number to multiply 6 to get 12.
Use the Equivalent Fractions Property to convert each fraction to an equivalent fraction with the LCD, multiplying both the numerator and denominator of each fraction by the same number.
Simplify the numerators and denominators.

We do not reduce the resulting fractions. If we did, we would get back to our original fractions and lose the common denominator.

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