# Add Fractions with Unlike Denominators – Step-by-step

Fractions with unlike denominators are fractions that have different denominators. To add these types of fractions, we have to find the least common denominator of the fractions. Then, we write equivalent fractions using the new denominator and add the like fractions (with the same denominator) normally.

Here, we will look at the steps that we can use to add fractions with different denominators. In addition, we will solve several practice problems to learn the concepts.

##### ARITHMETIC

Relevant for

Learning to add fractions with unlike denominators.

See steps

##### ARITHMETIC

Relevant for

Learning to add fractions with unlike denominators.

See steps

## Steps to add unlike fractions

We can add two or more fractions with different denominators by following these steps:

Step 1: Find the Least Common Denominator (LCD) of the fractions.

Step 2: Divide the LCD by the denominator of each fraction.

Step 3: Multiply both the numerator and the denominator by the result in step 2.

Step 4: Add the homogeneous fractions obtained from step 3.

Step 5: Simplify the final fraction if possible.

The following examples are solved using the steps for adding unlike fractions seen above. Try to solve the problems yourself before looking at the solution.

### EXAMPLE 1

What is the result of $latex \frac{1}{2}+\frac{1}{3}$?

Step 1: The least common denominator of 2 and 3 is 6.

Step 2: Dividing 6 by 2 (denominator of the first fraction), we get 3. Dividing 6 by 3 (denominator of the second fraction), we get 2.

Step 3: Multiplying the numerator and denominator of each fraction by the numbers obtained in step 2, we have:

$$\frac{1\times 3}{2 \times 3}+\frac{1 \times 2}{3 \times 2}$$

$$=\frac{3}{6}+\frac{2}{6}$$

Step 4: Adding the fractions from step 3, we have:

$$\frac{3}{6}+\frac{2}{6}$$

$$=\frac{3+2}{6}$$

$$=\frac{5}{6}$$

Step 5: The fraction is now simplified.

### EXAMPLE 2

Solve the addition of fractions $latex \frac{2}{3}+\frac{1}{4}$.

Step 1: The LCD of 3 and 4 is 12.

Step 2: Dividing 12 by 3 (denominator of the first fraction), we get 4. Dividing 12 by 4 (denominator of the second fraction), we get 3.

Step 3: By multiplying both the numerator and denominator of each fraction by the numbers obtained in step 2, we have:

$$\frac{2\times 4}{3 \times 4}+\frac{1 \times 3}{4 \times 3}$$

$$=\frac{8}{12}+\frac{3}{12}$$

Step 4: Adding the like fractions from step 3, we have:

$$\frac{8}{12}+\frac{3}{12}$$

$$=\frac{8+3}{12}$$

$$=\frac{11}{12}$$

Step 5: The fraction is now simplified.

### EXAMPLE 3

Find the result of the addition $latex \frac{3}{4}+\frac{2}{5}$.

Step 1: The least common denominator of 4 and 5 is 20.

Step 2: Dividing 20 by 4 (denominator of the first fraction), we get 5. Dividing 20 by 5 (denominator of the second fraction), we get 4.

Step 3: We multiply the numerators and denominators by the numbers obtained in step 2:

$$\frac{3\times 5}{4 \times 5}+\frac{2 \times 4}{5 \times 4}$$

$$=\frac{15}{20}+\frac{8}{20}$$

Step 4: Solving the addition of like fractions from step 3, we have:

$$\frac{15}{20}+\frac{8}{20}$$

$$=\frac{15+8}{20}$$

$$=\frac{23}{20}$$

Step 5: We can write the fraction as a mixed number:

$$=1\frac{3}{20}$$

### EXAMPLE 4

Find the result of $latex \frac{1}{3}+\frac{1}{4}+\frac{1}{2}$.

Step 1: The least common denominator of 3, 4, and 2 is 12.

Step 2: Dividing 12 by 3 (first denominator), we get 4. Dividing 12 by 4 (second denominator), we get 3. Dividing 12 by 2 (third denominator), we get 6.

Step 3: We multiply the numerators and denominators of each fraction by the numbers obtained in step 2:

$$\frac{1\times 4}{3 \times 4}+\frac{1 \times 3}{4 \times 3}+\frac{1 \times 6}{2 \times 6}$$

$$=\frac{4}{12}+\frac{3}{12}+\frac{6}{12}$$

Step 4: Solving the addition of like fractions from step 3, we have:

$$\frac{4}{12}+\frac{3}{12}+\frac{6}{12}$$

$$=\frac{4+3+6}{12}$$

$$=\frac{13}{12}$$

Step 5: Writing as a mixed number, we have:

$$=1\frac{1}{12}$$

### EXAMPLE 5

Solve the addition $latex \frac{2}{5}+\frac{3}{4}+\frac{1}{2}$.

Step 1: The least common denominator of 5, 4, and 2 is 20.

Step 2: Dividing 20 by 5 (first denominator), we get 4. Dividing 20 by 4 (second denominator), we get 5. Dividing 20 by 2 (third denominator), we get 10.

Step 3: By multiplying both the numerator and denominator of each fraction by the numbers obtained in step 2, we have:

$$\frac{2\times 4}{5 \times 4}+\frac{3 \times 5}{4 \times 5}+\frac{1 \times 10}{2 \times 10}$$

$$=\frac{8}{20}+\frac{15}{20}+\frac{10}{20}$$

Step 4: Solving the addition of fractions obtained in step 3, we have:

$$\frac{8}{20}+\frac{15}{20}+\frac{10}{20}$$

$$=\frac{8+15+10}{20}$$

$$=\frac{33}{20}$$

Step 5: Writing as a mixed number, we have:

$$=1\frac{13}{20}$$

### EXAMPLE 6

Solve the addition of fractions $latex \frac{3}{4}+\frac{2}{3}+\frac{4}{5}+\frac{1}{2}$.

Step 1: The least common denominator of 4, 3, 5, and 2 is 60.

Step 2: Dividing 60 by 4 (first denominator), we get 15. Dividing 60 by 3 (second denominator), we get 20. Dividing 60 by 5 (third denominator), we get 12. Dividing 60 by 2, we have 30.

Step 3: We multiply the numerators and denominators of each fraction by the numbers obtained in step 2:

$$\frac{3\times 15}{4 \times 15}+\frac{2 \times 20}{3 \times 20}+\frac{4 \times 12}{5 \times 12}+\frac{1 \times 30}{2 \times 30}$$

$$=\frac{45}{60}+\frac{40}{60}+\frac{48}{60}+\frac{30}{60}$$

Step 4: Solving the addition from step 3, we have:

$$\frac{45}{60}+\frac{40}{60}+\frac{48}{60}+\frac{30}{60}$$

$$=\frac{45+40+48+30}{60}$$

$$=\frac{163}{60}$$

Step 5: Writing as a mixed number, we have:

$$=2\frac{43}{60}$$

## Adding unlike fractions – Practice problems

Solve the following problems applying the process used to solve an addition of unlike fractions.