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# Least Common Denominators (LCDs)

Now that you've learned to add and subtract fractions with like denominators, it's time to learn how to do so when the denominator is not the same. We do this by using a process called finding the least common denominator (LCD). The least common denominator is the least common multiple (LCM) of the two denominators.

## Finding the least common denominator of fractions with low denominators

Example 1:

Find the least common denominator of the fractions $\frac{1}{6}$ and $\frac{3}{8}$ .

To find the common denominator of $\frac{1}{6}$ and $\frac{3}{8}$ , we need to find the least common multiple of 6 and 8. We can do this by listing the multiples and then finding the first one that is common to both numbers.

$6,12,18,24,30,36,42,48,54$

$8,16,18,24,32,36,40,48,56$

The first number that is a multiple of both 6 and 8 is 24. Therefore, 24 is the LCM, so that's what we use as our common denominator.

## Finding the least common denominator of fractions with higher denominators

Listing multiples can be impractical for larger numbers. When you run into this situation, it's best to use find their LCM by dividing their product by the greatest common factor (GCF).

Example 2:

Find the least common denominator of the fractions $\frac{5}{12}$ and $\frac{2}{15}$ .

First, find the greatest common factor of 12 and 15.

$12=3×2×2$

$15=5×3$

3 is the largest shared factor so it is the GCF.

Then, to find the least common multiple, divide the product of 12 and 15 by 3.

$\frac{12×15}{3}=\frac{3×4×5×3}{3}=60$

So the least common denominator is 60.

## Use least common denominators to add fractions

Once you have found the least common denominator in a pair of fractions, you can easily rewrite the problem using equivalent fractions. This makes them easy to add or subtract.

Example 3:

Add the fractions $\frac{5}{12}$ and $\frac{2}{15}$ .

We already figured out in the previous example that the least common denominator of these two fractions is 60.

To add these two fractions, you must write each one as an equivalent fraction in which the denominator is 60. To do this, we first must figure out how many times each denominator goes into 60.

$\frac{60}{12}=5$

$\frac{60}{15}=4$

Next, we take each fraction and multiply both the numerator and the denominator by the relevant quotient.

$\frac{5}{12}=\frac{\frac{5}{12}}{×}=\frac{25}{60}$

$\frac{2}{12}=\frac{2}{12}×\frac{4}{4}=\frac{8}{60}$

As you can see, this has the same effect as multiplying the fraction by 1 ( $\frac{5}{5}$ and $\frac{4}{4}$ are both = 1), so it doesn't change the value of the fraction at all.

Finally, you can add the fractions with the common denominators.

$\frac{5}{12}+\frac{2}{15}=\frac{25}{60}+\frac{8}{60}=\frac{33}{60}$

It's worth noting that this method does not always provide answers in the lowest terms, so you might still have some work. For example, both 33 and 60 are divisible by 3, so the usual way to write the answer is a little bit different.

$\frac{33}{60}=\frac{11}{30}$

## Add or subtract rational expressions

The same steps can be used to add or subtract fractions with variables.

Example 4:

Subtract $\frac{1}{2a}-\frac{1}{3b}$ for a and b not equal to 0.

Since the two expressions $2a$ and $3b$ have no common factors other than 1, their least common multiple is simply their product: $2a×3b$ .

Rewrite the two fractions using 6ab as the denominator.

$\frac{1}{2a}×\frac{3b}{3b}=\frac{3b}{6ab}$

$\frac{1}{3b}×\frac{2a}{2a}=\frac{2a}{6ab}$

Then subtract.

$\frac{1}{2a}-\frac{1}{3b}=\frac{3b}{6ab}-\frac{2a}{6ab}=\frac{3b-2a}{6ab}$

It doesn't matter where the variables are. The same principles apply.

Example 5:

Subtract $\frac{x}{16}-\frac{3}{8x}$ for x not equal to 0.

Since 16 and 8x have a common factor of 8, we find the least common multiple by dividing the product by 8.

$16×\frac{8x}{8}=16x$

The LCM is 16x so you multiply the first expression by 1 in the form of $\frac{x}{x}$ and the second expression by 1 in the form of $\frac{2}{2}$ .

$\frac{x}{16}×\frac{x}{x}=\frac{{x}^{2}}{16x}$

$\frac{3x}{8}×\frac{2}{2}=\frac{6}{16x}$

Subtract.

$\frac{x}{16}-\frac{3}{8x}=\frac{{x}^{2}}{16x}-\frac{6}{16x}=\frac{{x}^{2}-6}{16x}$

## Practice questions on finding Least Common Denominators

1. Find the least common denominator of the fractions $\frac{4}{7}$ and $\frac{2}{3}$ .

$3,6,9,12,15,18,21,24,27,30$

$7,14,21,28$

The least common denominator is 21.

2. Find the least common denominator of the fractions $\frac{5}{16}$ and $\frac{9}{12}$ .

The greatest common factor is 4.

$\frac{16×12}{4}=\frac{4×4×12}{4}=48$

So the least common denominator is 48.

3. Add the fractions $\frac{5}{16}$ and $\frac{9}{12}$ .

$\frac{48}{16}=3$

$\frac{48}{12}=4$

$\frac{5}{16}×\frac{3}{3}=\frac{15}{48}$

$\frac{9}{12}×\frac{4}{4}=\frac{36}{48}$

$\frac{5}{16}+\frac{9}{12}=\frac{15}{48}+\frac{36}{48}=\frac{51}{48}=\frac{17}{16}$

4. Subtract $\frac{1}{4}a-\frac{1}{3}b$ where a and b are not equal to 0.

The least common denominator is $12ab$ .

$\frac{1}{4}a×\frac{3b}{3b}=\frac{3b}{12ab}$

$\frac{1}{3}b×\frac{4a}{4a}=\frac{4a}{12ab}$

Then subtract.

$\frac{1}{4}a-\frac{1}{3}b=\frac{3b}{12ab}-\frac{4a}{12ab}=\frac{3b-4a}{12ab}$

5. Subtract $\frac{x}{14}-\frac{5}{7x}$ .

Find the least common multiple.

$14×\frac{7x}{7}=14x$

$\frac{x}{14}×\frac{x}{x}=\frac{{x}^{2}}{14x}$

$\frac{5}{7x}×\frac{2}{2}=\frac{10}{14x}$

Subtract.

$\frac{x}{14}-\frac{5}{4x}=\frac{{x}^{2}}{14x}-\frac{10}{14x}=\frac{{x}^{2}-10}{14x}$

## Get help learning about least common denominators

As you can see, finding the least common denominator can require some pretty serious calculations, especially for younger students who may not be used to working with formulas including variables. Working with a tutor can help your student get a handle on least common denominators and working with them in addition and subtraction of all kinds of fractions. To learn more about what tutoring can do for your student, contact the Educational Directors at Varsity Tutors today.

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