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

Fractions that have different denominators are called unlike fractions. We can subtract unlike fractions by finding the least common denominator and writing equivalent fractions using the new denominator. Then, we can subtract the like fractions (with the same denominator) obtained easily.

Here, we will learn to subtract heterogeneous fractions step by step. Then, we will look at some practice problems to apply the concepts learned.

##### ARITHMETIC

Relevant for

Learning to subtract fractions with unlike denominators.

See steps

##### ARITHMETIC

Relevant for

Learning to subtract fractions with unlike denominators.

See steps

## Steps to subtract unlike fractions

Unlike fractions are characterized by having different denominators. To subtract this type of fractions, we can follow the following steps:

Step 1: Calculate 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 of step 2. With this, we will obtain like fractions, where the LCD is the denominator.

Step 4: Add the like fractions obtained from step 3. For this, we use a single denominator and add the numerators.

Step 5: Simplify the resulting fraction if possible.

## Subtract unlike fractions – Examples with answers

The following examples of subtraction of unlike fractions have step-by-step solutions, but try to solve the problems yourself before looking at the answer.

### EXAMPLE 1

Find the result of the subtraction $latex \frac{1}{2}-\frac{1}{3}$.

Step 1: We have denominators 2 and 3. Therefore, the least common denominator is 6.

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

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

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

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

Step 4: Now we subtract the like fractions:

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

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

$$=\frac{1}{6}$$

Step 5: The fraction is now simplified.

### EXAMPLE 2

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

Step 1: We have denominators 3 and 4. Therefore, the LCD is 12.

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

Step 3: We multiply both the numerator and the denominator of each fraction by the numbers obtained in step 2, 4 for the first fraction, and 3 for the second:

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

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

Step 4: Subtracting the like fractions, we have:

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

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

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

Step 5: The fraction is now simplified.

### EXAMPLE 3

Solve the subtraction of fractions $latex \frac{3}{4}-\frac{2}{5}$.

Step 1: The denominators are 4 and 5, so the least common denominator is 20.

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

Step 3: We multiply the numerators and denominators by the numbers obtained in step 2, 5 for the first fraction, and 4 for the second:

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

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

Step 4: Solving the subtraction of like fractions, we have:

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

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

$$=\frac{7}{20}$$

Step 5: The fraction is now simplified.

### EXAMPLE 4

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

Step 1: We have denominators 2, 4, and 3, so the least common denominator is 12.

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

Step 3: We multiply the numerators and denominators of each fraction by the numbers obtained in step 2, 6 for the first fraction, 3 for the second and 4 for the third:

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

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

Step 4: Subtracting the like fractions, we have:

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

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

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

Step 5: Simplifying, we have:

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

### EXAMPLE 5

Find the result of the subtraction of fractions $latex \frac{7}{5}-\frac{3}{4}-\frac{1}{2}$.

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: Multiplying both the numerator and the denominator of each fraction by the numbers obtained in step 2, 4 for the first fraction, 5 for the second, and 10 for the third, we have:

$$\frac{7\times 4}{5 \times 4}-\frac{3 \times 5}{4 \times 5}-\frac{1 \times 10}{2 \times 10}$$

$$=\frac{28}{20}-\frac{15}{20}-\frac{10}{20}$$

Step 4: Solving the subtraction of like fractions, we have:

$$\frac{28}{20}-\frac{15}{20}-\frac{10}{20}$$

$$=\frac{28-15-10}{20}$$

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

Step 5: The fraction is now simplified.

### EXAMPLE 6

Solve the subtraction of fractions $latex \frac{10}{4}-\frac{2}{3}-\frac{4}{5}-\frac{1}{2}$.

Step 1: We have the denominators 4, 3, 5, and 2. The 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:

$$\frac{10\times 15}{4 \times 15}-\frac{2 \times 20}{3 \times 20}-\frac{4 \times 12}{5 \times 12}-\frac{1 \times 30}{2 \times 30}$$

$$=\frac{150}{60}-\frac{40}{60}-\frac{48}{60}-\frac{30}{60}$$

Step 4: Solving the subtraction of like fractions, we have:

$$\frac{150}{60}-\frac{40}{60}-\frac{48}{60}-\frac{30}{60}$$

$$=\frac{150-40-48-30}{60}$$

$$=\frac{32}{60}$$

Step 5: Simplifying the fraction, we have:

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

## Subtraction of unlike fractions – Practice problems

Use the process for subtracting unlike fractions above to solve the following problems.