# Adding Three or More Fractions – Step-by-step

To add 3 or more like fractions, we simply have to use a single denominator and add the numerators. On the other hand, to add 3 or more unlike fractions, we have to find the least common denominator and write equivalent fractions with that denominator. Then, we combine the fractions and add the numerators.

Here, we will learn how to add 3 or more fractions, both like fractions and unlike. We will use several examples with answers to understand the concepts.

##### ARITHMETIC

Relevant for

Learning to add 3 or more like and unlike fractions.

See steps

##### ARITHMETIC

Relevant for

Learning to add 3 or more like and unlike fractions.

See steps

## Steps to add three or more fractions

An addition of three or more fractions can be solved using the same steps used to solve the addition of two fractions. Depending on the type of fractions we have, we can use different steps.

### Adding three or more like fractions

To solve an addition of three or more fractions with the same denominators (like fractions), we can follow these steps:

Step 1: Recognize the numerator (top number) and denominator (bottom number) and make sure that the denominator of all fractions is the same.

Step 2: Use a single denominator to write the fractions. We can combine the fractions using a single denominator and form an addition with the numerators.

Step 3: Solve the addition of the numerators of the fraction obtained in step 2.

Step 4: Simplify the final fraction if possible.

### Adding three or more unlike fractions

To solve an addition of three or more fractions with different denominators (unlike fractions), we can follow 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 number obtained in step 2.

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

Step 5: Simplify the final fraction if possible.

## Adding 3 or more fractions – Examples with answers

The following examples are solved using the steps to add like and unlike fractions seen above. Try to solve the examples yourself before looking at the solution.

### EXAMPLE 1

Find the answer to the addition $latex \frac{3}{5}+\frac{2}{5}+\frac{1}{5}$.

Step 1: All three fractions have denominators equal to 5, so we have like fractions.

Step 2: Combining the fractions into a single denominator, we have:

$$\frac{3}{5}+\frac{2}{5}+\frac{1}{5}$$

$$=\frac{3+2+1}{5}$$

Step 3: Adding the numerators, we have:

$$=\frac{3+2+1}{5}$$

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

Step 4: Writing in mixed numbers, we have:

$$=1\frac{1}{5}$$

### EXAMPLE 2

Solve the addition $latex \frac{1}{5}+\frac{6}{10}+\frac{4}{5}$.

Step 1: The fractions have denominators 5, 10, and 5. These fractions do not appear to be like fractions at first glance. However, we can simplify the second fraction as follows:

$$\frac{1}{5}+\frac{6}{10}+\frac{4}{5}$$

$$=\frac{1}{5}+\frac{3}{5}+\frac{4}{5}$$

Step 2: Now, we can write as follows:

$$\frac{1}{5}+\frac{3}{5}+\frac{4}{5}$$

$$=\frac{1+3+4}{5}$$

Step 3: Adding the numerators, we have:

$$=\frac{1+3+4}{5}$$

$$=\frac{8}{5}$$

Step 4: We can write the fraction as a mixed number:

$$=1\frac{3}{5}$$

### EXAMPLE 3

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

In this case, we have unlike denominators, so we solve as follows:

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: By multiplying the numerators and denominators of each fraction by the numbers obtained in step 2, we have:

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

$$=\frac{8}{12}+\frac{3}{12}+\frac{6}{12}$$

Step 4: We solve the addition of like fractions from step 3:

$$\frac{8}{12}+\frac{3}{12}+\frac{6}{12}$$

$$=\frac{8+3+6}{12}$$

$$=\frac{17}{12}$$

Step 5: We can write as a mixed number:

$$=1\frac{5}{12}$$

### EXAMPLE 4

Find the result of $latex \frac{2}{5}+\frac{3}{4}+\frac{1}{2}$.

Step 1: We have the denominators 5, 4, and 2. The least common denominator 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: If we multiply both the numerator and the 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: Now, we solve the addition of like fractions from step 3, and 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 5

Find the result of $latex \frac{2}{3}+\frac{1}{3}+\frac{2}{7}+\frac{3}{7}$.

Step 1: The first two fractions have a denominator equal to 3 and the last two fractions have a denominator equal to 7. Therefore, the least common denominator is 21.

Step 2: Dividing 21 by 3 (first and second denominators), we get 7. Dividing 21 by 7 (third and fourth denominators), we get 3.

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

$$\frac{2\times 7}{3 \times 7}+\frac{1 \times 7}{3 \times 7}+\frac{2 \times 3}{7 \times 3}+\frac{3 \times 3}{7 \times 3}$$

$$=\frac{14}{21}+\frac{7}{21}+\frac{6}{21}+\frac{9}{21}$$

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

$$\frac{14}{21}+\frac{7}{21}+\frac{6}{21}+\frac{9}{21}$$

$$=\frac{14+7+6+9}{21}$$

$$=\frac{36}{21}$$

Step 5: Simplifying and writing as a mixed number, we have:

$$=\frac{12}{7}$$

$$=1\frac{5}{7}$$

### EXAMPLE 6

Find the result of $latex \frac{3}{4}+\frac{2}{3}+\frac{4}{5}+\frac{1}{2}$.

Step 1: We have the denominators 4, 3, 5 and 2. Their least common denominator 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 get 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 of like fractions 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:

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

## Addition of 3 or more fractions – Practice problems

Solve the following examples by applying everything learned about the addition of 3 or more like and unlike fractions.

#### What is the result of the addition $latex \frac{1}{7}+\frac{1}{2}+\frac{2}{3}+\frac{1}{3}$?  