Converting Fractions to Equivalent Fractions With the LCD

Learning Outcomes

  • Find the least common denominator of a group of fractions
  • Convert fractions to equivalent fractions using the least common denominator

In the previous section, we explained how to add and subtract fractions with a common denominator. But how can we add and subtract fractions with unlike denominators?

Let’s think about coins again. Can you add one quarter and one dime? You could say there are two coins, but that’s not very useful. To find the total value of one quarter plus one dime, you change them to the same kind of unit—cents. One quarter equals [latex]25[/latex] cents and one dime equals [latex]10[/latex] cents, so the sum is [latex]35[/latex] cents. See the image below.

Together, a quarter and a dime are worth [latex]35[/latex] cents, or [latex]{\Large\frac{35}{100}}[/latex] of a dollar.

A quarter and a dime are shown. Below them, it reads 25 cents plus 10 cents. Below that, it reads 35 cents.
Similarly, when we add fractions with different denominators we have to convert them to equivalent fractions with a common denominator. With the coins, when we convert to cents, the denominator is [latex]100[/latex]. Since there are [latex]100[/latex] cents in one dollar, [latex]25[/latex] cents is [latex]\Large\frac{25}{100}[/latex] and [latex]10[/latex] cents is [latex]\Large\frac{10}{100}[/latex]. So we add [latex]\Large\frac{25}{100}+\Large\frac{10}{100}[/latex] to get [latex]\Large\frac{35}{100}[/latex], which is [latex]35[/latex] cents.

You have practiced adding and subtracting fractions with common denominators. Now let’s see what you need to do with fractions that have different denominators.

First, we will use fraction tiles to model finding the common denominator of [latex]\Large\frac{1}{2}[/latex] and [latex]\Large\frac{1}{3}[/latex].

We’ll start with one [latex]\Large\frac{1}{2}[/latex] tile and [latex]\Large\frac{1}{3}[/latex] tile. We want to find a common fraction tile that we can use to match both [latex]\Large\frac{1}{2}[/latex] and [latex]\Large\frac{1}{3}[/latex] exactly.
If we try the [latex]\Large\frac{1}{4}[/latex] pieces, [latex]2[/latex] of them exactly match the [latex]\Large\frac{1}{2}[/latex] piece, but they do not exactly match the [latex]\Large\frac{1}{3}[/latex] piece.

Two rectangles are shown side by side. The first is labeled 1 half. The second is shorter and is labeled 1 third. Underneath the first rectangle is an equally sized rectangle split vertically into two pieces, each labeled 1 fourth. Underneath the second rectangle are two pieces, each labeled 1 fourth. These rectangles together are longer than the rectangle labeled as 1 third.
If we try the [latex]\Large\frac{1}{5}[/latex] pieces, they do not exactly cover the [latex]\Large\frac{1}{2}[/latex] piece or the [latex]\Large\frac{1}{3}[/latex] piece.

Two rectangles are shown side by side. The first is labeled 1 half. The second is shorter and is labeled 1 third. Underneath the first rectangle is an equally sized rectangle split vertically into three pieces, each labeled 1 sixth. Underneath the second rectangle is an equally sized rectangle split vertically into 2 pieces, each labeled 1 sixth.
If we try the [latex]\Large\frac{1}{6}[/latex] pieces, we see that exactly [latex]3[/latex] of them cover the [latex]\Large\frac{1}{2}[/latex] piece, and exactly [latex]2[/latex] of them cover the [latex]\Large\frac{1}{3}[/latex] piece.

Two rectangles are shown side by side. The first is labeled 1 half. The second is shorter and is labeled 1 third. Underneath the first rectangle are three smaller rectangles, each labeled 1 fifth. Together, these rectangles are longer than the 1 half rectangle. Below the 1 third rectangle are two smaller rectangles, each labeled 1 fifth. Together, these rectangles are longer than the 1 third rectangle.
If we were to try the [latex]\Large\frac{1}{12}[/latex] pieces, they would also work.

Two rectangles are shown side by side. The first is labeled 1 half. The second is shorter and is labeled 1 third. Underneath the first rectangle is an equally sized rectangle split vertically into 6 pieces, each labeled 1 twelfth. Underneath the second rectangle is an equally sized rectangle split vertically into 4 pieces, each labeled 1 twelfth.
Even smaller tiles, such as [latex]\Large\frac{1}{24}[/latex] and [latex]\Large\frac{1}{48}[/latex], would also exactly cover the [latex]\Large\frac{1}{2}[/latex] piece and the [latex]\Large\frac{1}{3}[/latex] piece.

The denominator of the largest piece that covers both fractions is the least common denominator (LCD) of the two fractions. So, the least common denominator of [latex]\Large\frac{1}{2}[/latex] and [latex]\Large\frac{1}{3}[/latex] is [latex]6[/latex].

Notice that all of the tiles that cover [latex]\Large\frac{1}{2}[/latex] and [latex]\Large\frac{1}{3}[/latex] have something in common: Their denominators are common multiples of [latex]2[/latex] and [latex]3[/latex], the denominators of [latex]\Large\frac{1}{2}[/latex] and [latex]\Large\frac{1}{3}[/latex]. The least common multiple (LCM) of the denominators is [latex]6[/latex], and so we say that [latex]6[/latex] is the least common denominator (LCD) of the fractions [latex]\Large\frac{1}{2}[/latex] and [latex]\Large\frac{1}{3}[/latex].

Least Common Denominator

The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators.

To find the LCD of two fractions, we will find the LCM of their denominators. We follow the procedure we used earlier to find the LCM of two numbers. We only use the denominators of the fractions, not the numerators, when finding the LCD.

Example

Find the LCD for the fractions: [latex]\Large\frac{7}{12}[/latex] and [latex]\Large\frac{5}{18}[/latex]

Solution:

Factor each denominator into its primes. Two adjacent factor trees for 12 and 18. The factors of 12 are 3 and 4, and the factors of 4 are 2 and 2. Thus, 2, 2, and 3 are the prime factors of 12. The factors of 18 are 3 and 6. The factors of 6 are 2 and 3. Thus, the prime factors of 18 are 2, 3, and 3.
List the primes of [latex]12[/latex] and the primes of [latex]18[/latex] lining them up in columns when possible. The image shows the prime factorization of 12 written as the equation 12 equals 2 times 2 times 3. Below this equation is another showing the prime factorization of 18 written as the equation 18 equals 2 times 3 times 3. The two equations line up vertically at the equal symbol. The first 2 in the prime factorization of 12 aligns with the 2 in the prime factorization of 18. The second 2 in the prime factorization of 12 does not align with any factor from the prime factorization of 18. The 3 in the prime factorization of 12 aligns with the first 3 from the prime factorization of 18. The second 3 in the prime factorization of 18 does not have any factor aligned above from the prime factorization of 12. A horizontal line is drawn under the prime factorization of 18.
Bring down the columns. The image shows the prime factorization of 12 written as the equation 12 equals 2 times 2 times 3. Below this equation is another showing the prime factorization of 18 written as the equation 18 equals 2 times 3 times 3. The two equations line up vertically at the equal symbol. The first 2 in the prime factorization of 12 aligns with the 2 in the prime factorization of 18. The second 2 in the prime factorization of 12 does not align with any factor from the prime factorization of 18. The 3 in the prime factorization of 12 aligns with the first 3 from the prime factorization of 18. The second 3 in the prime factorization of 18 does not have any factor aligned above from the prime factorization of 12. A horizontal line is drawn under the prime factorization of 18. Below this line is the equation LCM equal to 2 times 2 times 3 times 3. Arrows are drawn down vertically from the prime factorization of 12 through the prime factorization of 18 ending at the LCM equation. The first arrow starts at the first 2 in the prime factorization of 12 and continues down through the 2 in the prime factorization of 18. Ending with the first 2 in the LCM. The second arrow starts at the second 2 in the prime factorization of 12 and points down to the second 2 in the LCM. The third arrow starts at the 3 in the prime factorization of 12 and continues down through the first 3 in the prime factorization of 18. Ending with the first 3 in the LCM. The last arrow starts at the second 3 in the prime factorization of 18 and points down to the second 3 in the LCM.
Multiply the factors. The product is the LCM. [latex]\text{LCM}=36[/latex]
The LCM of [latex]12[/latex] and [latex]18[/latex] is [latex]36[/latex], so the LCD of [latex]\Large\frac{7}{12}[/latex] and [latex]\Large\frac{5}{18}[/latex] is 36. LCD of [latex]\Large\frac{7}{12}[/latex] and [latex]\Large\frac{5}{18}[/latex] is 36.

 

Try it

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.

Find the least common denominator (LCD) of two fractions

  1. Factor each denominator into its primes.
  2. List the primes, matching primes in columns when possible.
  3. Bring down the columns.
  4. Multiply the factors. The product is the LCM of the denominators.
  5. The LCM of the denominators is the LCD of the fractions.

Example

Find the least common denominator for the fractions: [latex]\Large\frac{8}{15}[/latex] and [latex]\Large\frac{11}{24}[/latex]

Try It

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

[latex]\Large\frac{1}{4}=\Large\frac{3}{12}\normalsize\text{ and }\Large\frac{1}{6}=\Large\frac{2}{12}[/latex].

On the left is a rectangle labeled 1 fourth. Below it is an identical rectangle split vertically into 3 equal pieces, each labeled 1 twelfth. On the right is a rectangle labeled 1 sixth. Below it is an identical rectangle split vertically into 2 equal pieces, each labeled 1 twelfth.
We say that [latex]\Large\frac{1}{4}\normalsize\text{ and }\Large\frac{3}{12}[/latex] are equivalent fractions and also that [latex]\Large\frac{1}{6}\normalsize\text{ and }\Large\frac{2}{12}[/latex] 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.

Equivalent Fractions Property

If [latex]a,b,c[/latex] are whole numbers where [latex]b\ne 0,c\ne 0,\text{then}[/latex]

[latex]\Large\frac{a}{b}=\Large\frac{a\cdot c}{b\cdot c}\normalsize\text{ and }\Large\frac{a\cdot c}{b\cdot c}=\Large\frac{a}{b}[/latex]

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 [latex]\Large\frac{1}{4}\normalsize\text{ and }\Large\frac{1}{6}[/latex] to equivalent fractions with denominator [latex]12[/latex] without using models.

Example

Convert [latex]\Large\frac{1}{4}\normalsize\text{ and }\Large\frac{1}{6}[/latex] to equivalent fractions with denominator [latex]12[/latex], their LCD.

Solution:

Find the LCD. The LCD of [latex]\Large\frac{1}{4}[/latex] and [latex]\Large\frac{1}{6}[/latex] is [latex]12[/latex].
Find the number to multiply [latex]4[/latex] to get [latex]12[/latex]. [latex]4\cdot\color{red}{3}=12[/latex]
Find the number to multiply [latex]6[/latex] to get [latex]12[/latex]. [latex]6\cdot\color{red}{2}=12[/latex]
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. [latex]\Large\frac{1}{4}[/latex]      [latex]\Large\frac{1}{6}[/latex]

[latex]\Large\frac{1\cdot\color{red}{3}}{4\cdot\color{red}{3}}[/latex]      [latex]\Large\frac{1\cdot\color{red}{2}}{6\cdot\color{red}{2}}[/latex]
Simplify the numerators and denominators. [latex]\Large\frac{3}{12}[/latex]   [latex]\Large\frac{2}{12}[/latex]

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

Try it

Convert two fractions to equivalent fractions with their LCD as the common denominator

  1. Find the LCD.
  2. For each fraction, determine the number needed to multiply the denominator to get the LCD.
  3. Use the Equivalent Fractions Property to multiply both the numerator and denominator by the number you found in Step 2.
  4. Simplify the numerator and denominator.

Example

Convert [latex]\Large\frac{8}{15}[/latex] and [latex]\Large\frac{11}{24}[/latex] to equivalent fractions with denominator [latex]120[/latex], their LCD.

 

Try it

In our next video we show two more examples of how to use the column method to find the least common denominator of two fractions.