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Study Guides > Intermediate Algebra

Read: Adding and Subtracting Fractions

Learning Objectives

  • Find the common denominator of two or more fractions
  • Use the common denominator to add or subtract fractions
  • Simplify a fraction to its lowest terms

Introduction

Before we get started, here is some important terminology that will help you understand the concepts about working with fractions in this section.
  • product: the result of  multiplication
  • factor: something being multiplied - for  32=63 \cdot 2 = 6 , both 33 and 22 are factors of 66
  • numerator: the top part of a fraction - the numerator in the fraction 23\Large\frac{2}{3} is 22
  • denominator: the bottom part of a fraction - the denominator in the fraction 23\Large\frac{2}{3} is 33

Note About Instructions

Many different words are used by math textbooks and teachers to provide students with instructions on what they are to do with a given problem. For example, you may see instructions such as "Find" or "Simplify" in the example in this module. It is important to understand what these words mean so you can successfully work through the problems in this course. Here is a short list of the words you may see that can help you know how to work through the problems in this module.
Instruction Interpretation
Find Perform the indicated mathematical operations which may include addition, subtraction, multiplication, division.
Simplify 1) Perform the indicated mathematical operations including addition, subtraction, multiplication, division 2) Write a mathematical statement in smallest terms so there are no other mathematical operations that can be performed—often found in problems related to fractions and the order of operations
Evaluate 1) Perform the indicated mathematical operations including addition, subtraction, multiplication, division 2) Substitute a given value for a variable in an expression and then perform the indicated mathematical operations
Reduce Write a mathematical statement in smallest or lowest terms so there are no other mathematical operations that can be performed—often found in problems related to fractions or division

Adding Fractions

When you need to add or subtract fractions, you will need to first make sure that the fractions have the same denominator. The denominator tells you how many pieces the whole has been broken into, and the numerator tells you how many of those pieces you are using. The “parts of a whole” concept can be modeled with pizzas and pizza slices. For example, imagine a pizza is cut into 44 pieces, and someone takes 11 piece. Now, 14\Large\frac{1}{4} of the pizza is gone and 34\Large\frac{3}{4} remains. Note that both of these fractions have a denominator of 44, which refers to the number of slices the whole pizza has been cut into. What if you have another pizza that had been cut into 88 equal parts and 33 of those parts were gone, leaving 58\Large\frac{5}{8}? A pizza divided into four slices, with one slice missing. How can you describe the total amount of pizza that is left with one number rather than two different fractions? You need a common denominator, technically called the least common multiple. Remember that if a number is a multiple of another, you can divide them and have no remainder. One way to find the least common multiple of two or more numbers is to first multiply each by 1,2,3,41, 2, 3, 4, etc.  For example, find the least common multiple of 22 and 55.
First, list all the multiples of 22: Then list all the multiples of 5:
21=22\cdot 1 = 2 51=55\cdot 1 = 5
22=42\cdot 2 = 4 52=105\cdot 2 = 10
23=62\cdot 3 = 6 53=155\cdot 3 = 15
24=82\cdot 4 = 8 54=205\cdot 4 = 20
25=102\cdot 5 = 10 55=255\cdot 5 = 25
The smallest multiple they have in common will be the common denominator for the two!

Example

Describe the amount of pizza left using common terms.

Answer: Rewrite the fractions 34\Large\frac{3}{4} and 58\Large\frac{5}{8} as fractions with a least common denominator. Find the least common multiple of the denominators. This is the least common denominator. Multiples of 4:4,8,12,16,244: 4, \textbf{8},12,16, \textbf{24} Multiples of 8:8,16,248: \textbf{8},16, \textbf{24} The least common denominator is 88—the smallest multiple they have in common. Rewrite 34\Large\frac{3}{4} with a denominator of 88. You have to multiply both the top and bottom by 22 so you don't change the relationship between them.

3422=68\Large\frac{3}{4}\cdot\Large\frac{2}{2}=\Large\frac{6}{8}

We don't need to rewrite 58\Large\frac{5}{8} since it already has the common denominator.

Answer

Both 68\Large\frac{6}{8} and 58\Large\frac{5}{8} have the same denominator, and you can describe how much pizza is left with common terms.

To add fractions with unlike denominators, first rewrite them with like denominators. Then, you know what to do! The steps are shown below.

Adding Fractions with Unlike Denominators

  1. Find a common denominator.
  2. Rewrite each fraction using the common denominator.
  3. Now that the fractions have a common denominator, you can add the numerators.
  4. Simplify by canceling out all common factors in the numerator and denominator.

Simplifying a Fraction

Often, if the answer to a problem is a fraction, you will be asked to write it in lowest terms. This is a common convention used in mathematics, similar to starting a sentence with a capital letter and ending it with a period. In this course, we will not go into great detail about methods for reducing fractions because there are many. The process of simplifying a fraction is often called reducing the fraction. We can simplify by canceling (dividing) the common factors in a fraction's numerator and denominator.  We can do this because a fraction represents division. For example, to simplify 69\Large\frac{6}{9} you can rewrite 66and 99 using the smallest factors possible as follows:

69=2333\Large\frac{6}{9}=\Large\frac{2\cdot3}{3\cdot3}

Since there is a 33 in both the numerator and denominator, and fractions can be considered division, we can divide the 33 in the top by the 33 in the bottom to reduce to 11.

69=2333=213=23\Large\frac{6}{9}=\Large\frac{2\cdot\cancel{3}}{3\cdot\cancel{3}}=\Large\frac{2\cdot1}{3}=\Large\frac{2}{3}

Rewriting fractions with the smallest factors possible is often called prime factorization. In the next example you are shown how to add two fractions with different denominators, then simplify the answer.

Example

Add 23+15\Large\frac{2}{3}+\Large\frac{1}{5}. Simplify the answer.

Answer: Since the denominators are not alike, find a common denominator by multiplying the denominators.

35=153\cdot5=15

Rewrite each fraction with a denominator of 1515.

2355=10151533=315\begin{array}{c}\Large\frac{2}{3}\cdot\Large\frac{5}{5}=\Large\frac{10}{15}\\\\\Large\frac{1}{5}\cdot\Large\frac{3}{3}=\Large\frac{3}{15}\end{array}

Add the fractions by adding the numerators and keeping the denominator the same. Make sure the fraction cannot be simplified.

1015+315=1315\Large\frac{10}{15}+\Large\frac{3}{15}=\Large\frac{13}{15}

Answer

23+15=1315\Large\frac{2}{3}+\Large\frac{1}{5}=\Large\frac{13}{15}

You can find a common denominator by finding the common multiples of the denominators. The least common multiple is the easiest to use.

Example

Add 37+221\Large\frac{3}{7}+\Large\frac{2}{21}. Simplify the answer.

Answer: Since the denominators are not alike, find the least common denominator by finding the least common multiple (LCM) of 7 and 21. Multiples of 7:7,14,217: 7, 14, \textbf{21} Multiples of 21:2121:\textbf{21} Rewrite each fraction with a denominator of 2121.

3733=921221\begin{array}{c}\Large\frac{3}{7}\cdot\Large\frac{3}{3}=\Large\frac{9}{21}\\\\\Large\frac{2}{21}\end{array}

Add the fractions by adding the numerators and keeping the denominator the same. Make sure the fraction cannot be simplified.

921+221=1121\Large\frac{9}{21}+\Large\frac{2}{21}=\Large\frac{11}{21}

Answer

37+221=1121\Large\frac{3}{7}+\Large\frac{2}{21}=\Large\frac{11}{21}

In the following video you will see an example of how to add two fractions with different denominators. https://youtu.be/zV4q7j1-89I You can also add more than two fractions as long as you first find a common denominator for all of them. An example of a sum of three fractions is shown below. In this example, you will use the prime factorization method to find the LCM.

Think About It

Add 34+16+58\Large\frac{3}{4}+\Large\frac{1}{6}+\Large\frac{5}{8}.  Simplify the answer and write as a mixed number. What makes this example different than the previous ones? Use the box below to write down a few thoughts about how you would add three fractions with different denominators together. [practice-area rows="2"][/practice-area]

Answer: Since the denominators are not alike, find the least common denominator by finding the least common multiple (LCM) of 4,64, 6, and 88.

4=226=328=222LCM:  2223=244=2\cdot2\\6=3\cdot2\\8=2\cdot2\cdot2\\\text{LCM}:\,\,2\cdot2\cdot2\cdot3=24

Rewrite each fraction with a denominator of 2424.

3466=18241644=4245833=1524\begin{array}{c}\Large\frac{3}{4}\cdot\Large\frac{6}{6}=\Large\frac{18}{24}\\\\\Large\frac{1}{6}\cdot\Large\frac{4}{4}=\Large\frac{4}{24}\\\\\Large\frac{5}{8}\cdot\Large\frac{3}{3}=\Large\frac{15}{24}\end{array}

Add the fractions by adding the numerators and keeping the denominator the same.

1824+424+1524=3724\Large\frac{18}{24}+\Large\frac{4}{24}+\Large\frac{15}{24}=\Large\frac{37}{24}

Write the improper fraction as a mixed number and simplify the fraction.

3724=1  1324\Large\frac{37}{24}=\normalsize 1\,\,\Large\frac{13}{24}

Answer

34+16+58=11324\Large\frac{3}{4}+\Large\frac{1}{6}+\Large\frac{5}{8}=\normalsize 1\Large\frac{13}{24}

Subtracting Fractions

When you subtract fractions, you must think about whether they have a common denominator, just like with adding fractions. Below are some examples of subtracting fractions whose denominators are not alike.

Example

Subtract 1516\Large\frac{1}{5}-\Large\frac{1}{6}. Simplify the answer.

Answer: The fractions have unlike denominators, so you need to find a common denominator. Recall that a common denominator can be found by multiplying the two denominators together.

56=305\cdot6=30

Rewrite each fraction as an equivalent fraction with a denominator of 3030.

1566=6301655=530\begin{array}{c}\Large\frac{1}{5}\cdot\Large\frac{6}{6}=\Large\frac{6}{30}\\\\\Large\frac{1}{6}\cdot\Large\frac{5}{5}=\Large\frac{5}{30}\end{array}

Subtract the numerators. Simplify the answer if needed.

630530=130\Large\frac{6}{30}-\Large\frac{5}{30}=\Large\frac{1}{30}

Answer

1516=130\Large\frac{1}{5}-\Large\frac{1}{6}=\Large\frac{1}{30}

The example below shows how to use multiples to find the least common multiple, which will be the least common denominator.

Example

Subtract 5614\Large\frac{5}{6}-\Large\frac{1}{4}. Simplify the answer.

Answer: Find the least common multiple of the denominators—this is the least common denominator. Multiples of  6:6,12,18,246: 6, \textbf{12}, 18, 24 Multiples of  4:4,8,12,16,204: 4, 8, \textbf{12},16, 20 1212 is the least common multiple of 66 and 44. Rewrite each fraction with a denominator of 1212.

5622=10121433=312\begin{array}{c}\Large\frac{5}{6}\cdot\Large\frac{2}{2}=\Large\frac{10}{12}\\\\\Large\frac{1}{4}\cdot\Large\frac{3}{3}=\Large\frac{3}{12}\end{array}

Subtract the fractions. Simplify the answer if needed.

1012312=712\Large\frac{10}{12}-\Large\frac{3}{12}=\Large\frac{7}{12}

Answer

5614=712\Large\frac{5}{6}-\Large\frac{1}{4}=\Large\frac{7}{12}

In the following video you will see an example of how to subtract fractions with unlike denominators. https://youtu.be/RpHtOMjeI7g

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