We can solve a difference of 3 or more like fractions by combining the denominators and subtracting the numerators. On the other hand, to subtract 3 or more unlike fractions, we have to find the least common denominator to write equivalent fractions with that denominator. Then we can add the numerators and use the new denominator.
Here, we will learn how to solve subtraction of 3 or more fractions, with both like and unlike fractions. In addition, we will look at some examples to understand the concepts.
Steps to subtract three or more fractions
To subtract three or more fractions, we can use steps similar to those used to subtract two fractions. Depending on the type of fractions we have, we will use a different process.
Let’s recall that like fractions are fractions that have the same denominators. On the other hand, unlike fractions are fractions that have different denominators.
Subtracting three or more like fractions
To solve a subtraction of three or more like fractions, we follow these steps:
Step 1: Make sure that the denominator is the same in all fractions. In some cases, it is possible to obtain like fractions, after simplifying some of the fractions.
Step 2: Combine the fractions using a single denominator.
Step 3: Solve the subtraction of the numerators of the fraction obtained in step 2.
Step 4: Simplify the resulting fraction if possible.
Subtracting three or more unlike fractions
To solve a subtraction of three or more unlike fractions, we follow these steps:
Step 1: Find the least common denominator (LCD) of the fractions.
Step 2: Divide the least common denominator by the denominator of each fraction.
Step 3: Multiply both the numerator and the denominator of each fraction by the numbers obtained in step 2. With this, we will obtain like fractions.
Step 4: Solve the subtraction of like fractions obtained in step 3.
Step 5: Simplify the resulting fraction if possible.
Subtracting 3 or more fractions – Examples with answers
Each of the following examples has its respective step-by-step solution. The processes for solving subtraction of 3 or more fractions seen above are used to solve these problems.
EXAMPLE 1
Solve the subtraction of fractions $latex \frac{4}{5}-\frac{2}{5}-\frac{1}{5}$.
Solution
Step 1: The denominators of the three fractions are equal to 5, so the fractions are like.
Step 2: Using a single denominator to combine the fractions, we have:
$$\frac{4}{5}-\frac{2}{5}-\frac{1}{5}$$
$$=\frac{4-2-1}{5}$$
Step 3: Solving the subtraction in the numerators, we have:
$$=\frac{4-2-1}{5}$$
$$=\frac{1}{5}$$
Step 4: The fraction is now simplified.
EXAMPLE 2
Find the result of $latex \frac{9}{5}-\frac{6}{10}-\frac{4}{5}$.
Solution
Step 1: The fractions do not appear to be like at first glance. However, we can simplify the second fraction as follows:
$$\frac{9}{5}-\frac{6}{10}-\frac{4}{5}$$
$$=\frac{9}{5}-\frac{3}{5}-\frac{4}{5}$$
Step 2: Combining the denominators into one, we have:
$$\frac{9}{5}-\frac{3}{5}-\frac{4}{5}$$
$$=\frac{9-3-4}{5}$$
Step 3: Solving the subtraction of numerators, we have:
$$=\frac{9-3-4}{5}$$
$$=\frac{2}{5}$$
Step 4: The fraction is now simplified.
EXAMPLE 3
Solve the subtraction $latex \frac{4}{3}-\frac{1}{4}-\frac{1}{2}$.
Solution
We have a subtraction of unlike fractions, so we solve like this:
Step 1: The denominators are 3, 4, and 2. The least common denominator 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; 4 for the first fraction, 3 for the second, and 6 for the third:
$$\frac{4\times 4}{3 \times 4}-\frac{1 \times 3}{4 \times 3}-\frac{1 \times 6}{2 \times 6}$$
$$=\frac{16}{12}-\frac{3}{12}-\frac{6}{12}$$
Step 4: Solving the subtraction of like fractions, we have:
$$\frac{16}{12}-\frac{3}{12}-\frac{6}{12}$$
$$=\frac{16-3-6}{12}$$
$$=\frac{7}{12}$$
Step 5: The fraction is now simplified.
EXAMPLE 4
Solve the subtraction of fractions $latex \frac{7}{5}-\frac{3}{4}-\frac{1}{2}$.
Solution
Step 1: We have the denominators 5, 4, and 2. Therefore, 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: We multiply 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:
$$\frac{7\times 4}{5 \times 4}-\frac{3 \times 5}{4 \times 5}-\frac{1 \times 10}{2 \times 10}$$
$$=\frac{28}{20}-\frac{15}{20}-\frac{10}{20}$$
Step 4: Solving the subtraction, 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 5
Solve the following $latex \frac{2}{3}-\frac{1}{3}+\frac{2}{7}-\frac{3}{7}$.
Solution
Step 1: The first two denominators equal 3 and the last two denominators equal 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: We multiply both the numerators and the denominators of each fraction by the numbers obtained in step 2:
$$\frac{2\times 7}{3 \times 7}-\frac{1 \times 7}{3 \times 7}+\frac{2 \times 3}{7 \times 3}-\frac{3 \times 3}{7 \times 3}$$
$$=\frac{14}{21}-\frac{7}{21}+\frac{6}{21}-\frac{9}{21}$$
Step 4: Solving the addition and subtraction of like fractions, we have:
$$\frac{14}{21}-\frac{7}{21}+\frac{6}{21}-\frac{9}{21}$$
$$=\frac{14-7+6-9}{21}$$
$$=\frac{4}{21}$$
Step 5: The fraction is now simplified.
EXAMPLE 6
Solve the following $latex \frac{3}{4}-\frac{2}{3}+\frac{4}{5}-\frac{1}{2}$.
Solution
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: By multiplying the numerators and denominators of each fraction by the numbers obtained in step 2, we have
$$\frac{3\times 15}{4 \times 15}-\frac{2 \times 20}{3 \times 20}+\frac{4 \times 12}{5 \times 12}-\frac{1 \times 30}{2 \times 30}$$
$$=\frac{45}{60}-\frac{40}{60}+\frac{48}{60}-\frac{30}{60}$$
Step 4: Solving the additions and subtractions of like fractions, we have:
$$\frac{45}{60}-\frac{40}{60}+\frac{48}{60}-\frac{30}{60}$$
$$=\frac{45-40+48-30}{60}$$
$$=\frac{23}{60}$$
Step 5: The fraction is now simplified.
→ Subtracting Fractions Calculator
Subtraction of 3 or more fractions – Practice problems
Apply everything learned about subtraction of 3 or more like and unlike fractions to solve the following practice problems.
See also
Interested in learning more about subtracting fractions? You can take a look at these pages:
Jefferson Huera Guzman
Jefferson is the lead author and administrator of Neurochispas.com. The interactive Mathematics and Physics content that I have created has helped many students.
