Addition and Subtraction of Algebraic Fractions - Examples, Exercises and Solutions

The key to adding or subtracting algebraic fractions is to make all denominators equal, that is, to find the common denominator.
To do this, we will need to factorize according to the different methods we have learned.

Action steps:

  1. We will factorize all the denominators we have.
  2. We will note the common denominator and, in this way, we will know how to meticulously carry out the third step.
  3. We will multiply each of the numerators by the same number that we need to multiply its denominator in order to reach the common denominator.
  4. We will write the exercise with a single denominator, the common denominator, and among the numerators, we will keep the same mathematical operations that were in the original exercise.
  5. After opening the parentheses, it may happen that we encounter another expression that needs to be factorized. We will factorize it and see if we can simplify it.
  6. We will obtain a common fraction and solve it.
Detailed explanation

Suggested Topics to Practice in Advance

  1. Factoring using contracted multiplication
  2. Factorization
  3. Extracting the common factor in parentheses
  4. Factorization: Common factor extraction
  5. Factoring Trinomials

Practice Addition and Subtraction of Algebraic Fractions

Question 1

Determine if the simplification below is correct:

\( \frac{5\cdot8}{8\cdot3}=\frac{5}{3} \)

Question 2

Determine if the simplification shown below is correct:

\( \frac{7}{7\cdot8}=8 \)

Question 3

Determine if the simplification below is correct:

\( \frac{4\cdot8}{4}=\frac{1}{8} \)

Question 4

Determine if the simplification below is correct:

\( \frac{3\cdot7}{7\cdot3}=0 \)

Question 5

Determine if the simplification below is correct:

\( \frac{6\cdot3}{6\cdot3}=1 \)

examples with solutions for addition and subtraction of algebraic fractions

Exercise #1

Determine if the simplification below is correct:

5⋅88⋅3=53 \frac{5\cdot8}{8\cdot3}=\frac{5}{3}

Video Solution

Step-by-Step Solution

Let's consider the fraction and break it down into two multiplication exercises:

88×53 \frac{8}{8}\times\frac{5}{3}

We simplify:

1×53=53 1\times\frac{5}{3}=\frac{5}{3}

Answer

Correct

Exercise #2

Determine if the simplification shown below is correct:

77⋅8=8 \frac{7}{7\cdot8}=8

Video Solution

Step-by-Step Solution

Let's consider the fraction and break it down into two multiplication exercises:

77×18 \frac{7}{7}\times\frac{1}{8}

We simplify:

1×18=18 1\times\frac{1}{8}=\frac{1}{8}

Therefore, the described simplification is false.

Answer

Incorrect

Exercise #3

Determine if the simplification below is correct:

4⋅84=18 \frac{4\cdot8}{4}=\frac{1}{8}

Video Solution

Step-by-Step Solution

We will divide the fraction exercise into two multiplication exercises:

44×81= \frac{4}{4}\times\frac{8}{1}=

We simplify:

1×81=8 1\times\frac{8}{1}=8

Therefore, the described simplification is false.

Answer

Incorrect

Exercise #4

Determine if the simplification below is correct:

3⋅77⋅3=0 \frac{3\cdot7}{7\cdot3}=0

Video Solution

Step-by-Step Solution

We will divide the fraction exercise into two different multiplication exercises.

As this is a multiplication exercise, you can use the substitution property:

77×33=1×1=1 \frac{7}{7}\times\frac{3}{3}=1\times1=1

Therefore, the simplification described is false.

Answer

Incorrect

Exercise #5

Determine if the simplification below is correct:

6⋅36⋅3=1 \frac{6\cdot3}{6\cdot3}=1

Video Solution

Step-by-Step Solution

We simplify the expression on the left side of the approximate equality:

6̸⋅3̸6̸⋅3̸=?1↓1=!1 \frac{\textcolor{red}{\not{6}}\cdot\textcolor{blue}{\not{3}}}{\textcolor{red}{\not{6}}\cdot\textcolor{blue}{\not{3}}}\stackrel{?}{= }1\\ \downarrow\\ 1\stackrel{!}{= }1 therefore, the described simplification is correct.

Therefore, the correct answer is A.

Answer

Correct

Question 1

Complete the corresponding expression for the denominator

\( \frac{16ab}{?}=8a \)

Question 2

Determine if the simplification described below is correct:

\( \frac{x+6}{y+6}=\frac{x}{y} \)

Question 3

Determine if the simplification below is correct:

\( \frac{3-x}{-x+3}=0 \)

Question 4

Determine if the simplification below is correct:

\( \frac{3\cdot4}{8\cdot3}=\frac{1}{2} \)

Question 5

Select the field of application of the following fraction:

\( \frac{x}{16} \)

examples with solutions for addition and subtraction of algebraic fractions

Exercise #1

Complete the corresponding expression for the denominator

16ab?=8a \frac{16ab}{?}=8a

Video Solution

Step-by-Step Solution

We use the formula:

xy=zw→x⋅y=z⋅y \frac{x}{y}=\frac{z}{w}\xrightarrow{}x\cdot y=z\cdot y

We convert the 8 into a fraction, and multiply

16ab?=81 \frac{16ab}{?}=\frac{8}{1}

16ab×1=8a 16ab\times1=8a

16ab=8a 16ab=8a

We divide both sides by 8a:

16ab8a=8a8a \frac{16ab}{8a}=\frac{8a}{8a}

2b 2b

Answer

2b 2b

Exercise #2

Determine if the simplification described below is correct:

x+6y+6=xy \frac{x+6}{y+6}=\frac{x}{y}

Video Solution

Step-by-Step Solution

We use the formula:

x+zy+z=x+zy+z \frac{x+z}{y+z}=\frac{x+z}{y+z}

x+6y+6=x+6y+6 \frac{x+6}{y+6}=\frac{x+6}{y+6}

Therefore, the simplification described is incorrect.

Answer

Incorrect

Exercise #3

Determine if the simplification below is correct:

3−x−x+3=0 \frac{3-x}{-x+3}=0

Video Solution

Step-by-Step Solution

z−x−x+z=1 \frac{z-x}{-x+z}=1

Answer

Incorrect

Exercise #4

Determine if the simplification below is correct:

3⋅48⋅3=12 \frac{3\cdot4}{8\cdot3}=\frac{1}{2}

Video Solution

Step-by-Step Solution

We simplify the expression on the left side of the approximate equality.

First let's consider the fact that the number 8 is a multiple of the number 4:

8=2⋅4 8=2\cdot4
Therefore, we will return to the problem in question and present the number 8 as a multiple of the number 4, then we will simplify the fraction:

3⋅48‾⋅3=?12↓3⋅42⋅4‾⋅3=?12↓3̸⋅4̸2⋅4̸⋅3̸=?12↓12=!12 \frac{3\cdot4}{\underline{8}\cdot3}\stackrel{?}{= }\frac{1}{2}\\ \downarrow\\ \frac{3\cdot4}{\underline{2\cdot4}\cdot3}\stackrel{?}{= }\frac{1}{2}\\ \downarrow\\ \frac{\textcolor{blue}{\not{3}}\cdot\textcolor{red}{\not{4}}}{2\cdot\textcolor{red}{\not{4}}\cdot\textcolor{blue}{\not{3}}}\stackrel{?}{= }\frac{1}{2} \\ \downarrow\\ \frac{1}{2}\stackrel{!}{= }\frac{1}{2}
Therefore, the described simplification is correct.

That is, the correct answer is A.

Answer

True

Exercise #5

Select the field of application of the following fraction:

x16 \frac{x}{16}

Video Solution

Answer

All X All~X

Question 1

Select the field of application of the following fraction:

\( \frac{8+x}{5} \)

Question 2

Select the field of application of the following fraction:

\( \frac{6}{x} \)

Question 3

Select the field of application of the following fraction:

\( \frac{3}{x+2} \)

Question 4

Select the field of application of the following fraction:

\( \frac{8}{-2+x} \)

Question 5

Select the field of application of the following fraction:

\( \frac{7}{13+x} \)

examples with solutions for addition and subtraction of algebraic fractions

Exercise #1

Select the field of application of the following fraction:

8+x5 \frac{8+x}{5}

Video Solution

Answer

All numbers

Exercise #2

Select the field of application of the following fraction:

6x \frac{6}{x}

Video Solution

Answer

All numbers except 0

Exercise #3

Select the field of application of the following fraction:

3x+2 \frac{3}{x+2}

Video Solution

Answer

x≠−2 x\neq-2

Exercise #4

Select the field of application of the following fraction:

8−2+x \frac{8}{-2+x}

Video Solution

Answer

x≠2 x\neq2

Exercise #5

Select the field of application of the following fraction:

713+x \frac{7}{13+x}

Video Solution

Answer

x≠−13 x\neq-13

Topics learned in later sections

  1. Algebraic Fractions
  2. Simplifying Algebraic Fractions
  3. Factoring Algebraic Fractions
  4. Multiplication and Division of Algebraic Fractions
  5. Solving Equations by Factoring
  6. Uses of Factorization