Mathematics » Introducing Fractions » Add and Subtract Fractions with Different Denominators

Finding the Least Common Denominator

Finding the Least Common Denominator

In previous lessons, 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 \(25\) cents and one dime equals \(10\) cents, so the sum is \(35\) cents. See the figure below.

A quarter and a dime are shown. Below them, it reads 25 cents plus 10 cents. Below that, it reads 35 cents.

Together, a quarter and a dime are worth \(35\) cents, or \(\frac{35}{100}\) of a dollar.

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 \(100.\) Since there are \(100\) cents in one dollar, \(25\) cents is \(\frac{25}{100}\) and \(10\) cents is \(\frac{10}{100}.\) So we add \(\frac{25}{100}+\frac{10}{100}\) to get \(\frac{35}{100},\) which is \(35\) 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 \(\frac{1}{2}\) and \(\frac{1}{3}.\)

We’ll start with one \(\frac{1}{2}\) tile and \(\frac{1}{3}\) tile. We want to find a common fraction tile that we can use to match both \(\frac{1}{2}\) and \(\frac{1}{3}\) exactly.

If we try the \(\frac{1}{4}\) pieces, \(2\) of them exactly match the \(\frac{1}{2}\) piece, but they do not exactly match the \(\frac{1}{3}\) 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 \(\frac{1}{5}\) pieces, they do not exactly cover the \(\frac{1}{2}\) piece or the \(\frac{1}{3}\) 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 \(\frac{1}{6}\) pieces, we see that exactly \(3\) of them cover the \(\frac{1}{2}\) piece, and exactly \(2\) of them cover the \(\frac{1}{3}\) 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 \(\frac{1}{12}\) 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 \(\frac{1}{24}\) and \(\frac{1}{48},\) would also exactly cover the \(\frac{1}{2}\) piece and the \(\frac{1}{3}\) 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 \(\frac{1}{2}\) and \(\frac{1}{3}\) is \(6.\)

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

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 \(\frac{7}{12}\) 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.

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.

Finding the Least Common Denominator 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 \(\frac{8}{15}\) and \(\frac{11}{24}.\)

Solution

To find the LCD, we find the LCM of the denominators.

Find the LCM of \(15\) and \(24.\)

The top line shows 15 equals 3 times 5. The next line shows 24 equals 2 times 2 times 2 times 3. The 3s are lined up vertically. The next line shows LCM equals 2 times 2 times 2 times 3 times 5. The last line shows LCM equals 120.

The LCM of \(15\) and \(24\) is \(120.\) So, the LCD of \(\frac{8}{15}\) and \(\frac{11}{24}\) is \(120.\)


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