## Finding the Least Common Denominator

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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.

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.

If we try the \(\frac{1}{5}\) pieces, they do not exactly cover the \(\frac{1}{2}\) piece or the \(\frac{1}{3}\) piece.

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.

If we were to try the \(\frac{1}{12}\) pieces, they would also work.

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

- Factor each denominator into its primes.
- List the primes, matching primes in columns when possible.
- Bring down the columns.
- Multiply the factors. The product is the LCM of the denominators.
- 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 LCM of \(15\) and \(24\) is \(120.\) So, the LCD of \(\frac{8}{15}\) and \(\frac{11}{24}\) is \(120.\)