Lowest Common Denominator

The Lowest Common Denominator of \( \frac{2}{3}\) and \( \frac{3}{4}\) is \(12\), since \(12\) is a multiple of both the denominators \((3\) and \(4),\) and it's the smallest multiple of these numbers you can find.

For instance, the LCD between fractions $\frac{3}{2}$ and $\frac{1}{2}$ is 2 because the denominator of the fractions is the same.

The LCD between $\frac{3}{4}$, $\frac{1}{2}$, and $\frac{3}{20}$ is 20. This is because, in the denominators list of 2, 4 and 20, we have that 2 and 4 are factors of 20. Thus, 20 is a multiple of 2 and 4, so, 20 is the LCD.

The LCD between $\frac{2}{3}$ and $\frac{1}{5}$ is 15. These denominators 3 and 5 are prime numbers. Thus,

$3\times 5=15$

The LCD between $\frac{7}{9}$ and $\frac{3}{10}$ is 90. Denominators 9 and 10 are not prime numbers. Most importantly, there is no common factor between these two numbers. So the LCD is

$9\times 10=90$

The LCD between $\frac{3}{8}$ and $\frac{7}{20}$ is 40. The highest common factor between 8 and 20 is 4. Divide 8 by 4 and your first answer is 2. Then multiply your first result which is two by the other number 20,

$2\times 20=40$

So the LCD, in this case, is 40.

Find the LCD of $\frac{1}{2}$, $\frac{3}{4}$ and $\frac{5}{6}$.

Solution:

Step 1. What are the denominators?

The denominators are 2, 4 and 6

Step 2. Write out the multiple of these numbers.

Multiples of 2 are 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, ...

Multiples of 4 are 0, 4, 8, 12, 16, 20, 24 , 28, 32, 36, 40, 44, 48, ...

Multiples of 6 are 0, 6, 12, 18, 24 , 30, 36, 42, 48, 54, 60, 66, 72, ...

You would notice that 12 and 24 are common multiples of 2, 4 and 6. But, we want the lowest common multiple. So, our lowest common multiple is 12. This means that our LCD is 12 .

Find the LCD between $\frac{2}{5}$ and $\frac{2}{7}$

Solution:

Step 1. What are the denominators?

The denominators are 5 and 7

Step 2. Write out the multiple of these numbers.

Multiples of 5 are 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, ...

Multiples of 7 are 0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, ...

From the above, it is clear that the LCD is 35.

Find the LCD between$\frac{1}{2}$, $\frac{1}{3}$ and $\frac{3}{4}$.

Solution:

Step 1. What are the denominators?

The denominators are 2, 3 and 4

Step 2. Write out the multiple of these numbers.

Multiples of 2 are 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, ...

Multiples of 3 are 0, 3, 6, 9, 12, 15, 18, 21, 24 , 27, 30, 33, 36, ...

Multiples of 4 are 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, ...

You would notice that 12 and 24 are common multiples of 2, 4 and 6. But, we want the lowest common multiple. So, our lowest common multiple is 12. This means that our LCD is 12.

Find the LCD between $\frac{11}{24}$ and $\frac{7}{10}.$

Solution:

Step 1: Write out the denominators.

The denominators are; 24 and 10.

Step 2: Express each denominator as a product of its prime factor. To achieve this, you have to follow some simple sub-steps.

Sub-step A: Draw a column table with the denominator in the second column.

Sub-step B: Use the smallest factor of the denominator and divide the number gradually until you arrive at 1. Remember that the factor is written in the first column.

Sub-step C: Write all factors in the first column as a product.

Repeat the same step for the other denominator and you would have:

Step 3: Arrange these prime factors being multiplied and circle out similar factors.

Third step in the prime factor method of finding the LCD, StudySmarter Originals

This means that the common factor between 24 and 10 is 2.

Step 4: Multiply the rest of the factors which are not circled with your common factor.

So, our LCD is;

$\mathbf{2}\mathbf{\times }\mathbf{5}\mathbf{\times }\mathbf{2}\mathbf{\times }\mathbf{2}\mathbf{\times }\mathbf{3}\mathbf{=}\mathbf{120}$

Find the LCD between $\frac{11}{24}$ and $\frac{7}{10}.$

Solution:

Step 1. Write out the denominators.

The denominators are 24 and 10.

Step 2. Create a table with the number of columns according to the number of denominators. In this case, the number of columns is 3 (one more than the number of denominators). Leave the first column for the factors.

Step 3. Use the smallest prime factor to divide the denominators. Note that a factor may only be able to divide one denominator but fail to divide the rest. In such cases where the factor cannot divide one of the numbers in the denominator column, just write the same number in the cell below. You just need that factor to divide at least one of the denominators. It is easier by firstly trying out the smaller numbers like, 2,3, ... Remember that the factor is written in the first column.

Note that in this case 2 can divide both 24 and 10 without a remainder.

Note that 2 can divide 12 but cannot divide 5, so we still bring down 5 in the cell below.

Step 4: Write all factors in the first column as a product.

Once you are left with ones in the last row, you can calculate the LCD as the product of the factors in the first column.

The LCD between the denominators 24 and 10 is then

$2\times 2\times 2\times 3\times 5=120$

The combined prime factor product method is the best method for a large set of denominators.

Simplify

$\frac{1}{2}-\frac{1}{4}$

Solution:

Step 1. Find the LCD of the denominators.

$\frac{1}{2}-\frac{1}{4}=\frac{1\times (4÷2)-1\times (4÷4)}{4}$

$$\begin{gathered} \frac{1\times \left(2\right)-1\times \left(1\right)}{4}= \\ =\frac{2-1}{4} \\ =\frac{1}{4} \end{gathered}$$

Simplify

$\frac{3}{5}+\frac{5}{7}+\frac{1}{28}$

Solution:

Step 1: Find the LCD of the denominators.

Use the combined prime factor product method.

The LCD of $\frac{3}{5}$, $\frac{5}{7}$ and $\frac{1}{28}$ is the product of the numbers in the first column, that is

$2\times 2\times 5\times 7=140$.

Step 2: Use the LCD as a general denominator. Then divide the LCD by each denominator and multiply by the numerator.

$\frac{3}{5}+\frac{5}{7}+\frac{1}{28}=\frac{3\times (140÷5)+5\times (140÷7)+1\times (140÷28)}{140}$

Step 3: Solve the arithmetic.

$$\begin{gathered} \frac{3\times \left(28\right)+5\times \left(20\right)+1\times \left(5\right)}{140}= \\ =\frac{84+100+5}{140} \\ =\frac{189}{140} \end{gathered}$$

Step 4: See if any number can divide through the numerator and denominator to simplify your fraction.

Divide the numerator and denominator by 7

id="3047903" role="math" $$\begin{gathered} \frac{189÷7}{140÷7}=\frac{27}{20} \\ =1\frac{7}{20} \end{gathered}$$

The LCD of denominators 6, 20, 8 and 5 is

$2\times 2\times 2\times 3\times 5=120$.

Step 2: Use the LCD as a general denominator. Then divide the LCD by each denominator and multiply by the numerator separately.

$$\begin{gathered} \frac{1}{6}=\frac{1\times (120÷6)}{120}=\frac{1\times \left(20\right)}{120}=\frac{\mathbf{20}}{120} \\ \frac{3}{20}=\frac{3\times (120÷20)}{120}=\frac{3\times \left(6\right)}{120}=\frac{\mathbf{18}}{120} \\ \frac{1}{8}=\frac{1\times (120÷8)}{120}=\frac{1\times \left(15\right)}{120}=\frac{\mathbf{15}}{120} \\ \frac{2}{5}=\frac{2\times (120÷5)}{120}=\frac{2\times \left(24\right)}{120}=\frac{\mathbf{48}}{120} \end{gathered}$$

Step 3: Only use the numerators in bold. Now arrange them from highest to lowest.

48, 20, 18 and 15 which is

$\frac{\mathbf{2}}{\mathbf{5}}\mathbf{\text{'}}\mathbf{}\frac{\mathbf{1}}{\mathbf{6}}\mathbf{,}\mathbf{}\frac{\mathbf{3}}{\mathbf{20}}\mathbf{}$ and $\frac{\mathbf{1}}{\mathbf{8}}$