Adding and Subtracting Fractions
Learning Outcomes
- 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 [latex]3 \cdot 2 = 6[/latex] , both [latex]3[/latex] and [latex]2[/latex] are factors of [latex]6[/latex]
- numerator: the top part of a fraction — the numerator in the fraction [latex]\dfrac{2}{3}[/latex] is [latex]2[/latex]
- denominator: the bottom part of a fraction — the denominator in the fraction [latex]\dfrac{2}{3}[/latex] is [latex]3[/latex]
Math Question 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 [latex]4[/latex] pieces, and someone takes [latex]1[/latex] piece. Now, [latex]\dfrac{1}{4}[/latex] of the pizza is gone and [latex]\dfrac{3}{4}[/latex] remains.
Note that both of these fractions have a denominator of [latex]4[/latex], 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 [latex]8[/latex] equal parts and [latex]3[/latex] of those parts were gone, leaving [latex]\dfrac{5}{8}[/latex]?
How can you describe the total amount of pizza that is left between the two pizzas? There are now two very different fractions and it's hard to compare the two. You will 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 [latex]1, 2, 3, 4[/latex], etc. For example, find the least common multiple of [latex]2[/latex] and [latex]5[/latex].
| First, list all the multiples of [latex]2[/latex]: | Then list all the multiples of 5: |
| [latex]2\cdot 1 = 2[/latex] | [latex]5\cdot 1 = 5[/latex] |
| [latex]2\cdot 2 = 4[/latex] | [latex]5\cdot 2 = 10[/latex] |
| [latex]2\cdot 3 = 6[/latex] | [latex]5\cdot 3 = 15[/latex] |
| [latex]2\cdot 4 = 8[/latex] | [latex]5\cdot 4 = 20[/latex] |
| [latex]2\cdot 5 = 10[/latex] | [latex]5\cdot 5 = 25[/latex] |
Example
Describe the amount of pizza left in the examples above using common terms.Answer: Rewrite the fractions [latex]\dfrac{3}{4}[/latex] and [latex]\dfrac{5}{8}[/latex] as fractions with a least common denominator. Find the least common multiple of the denominators. This is the least common denominator. Multiples of [latex]4: 4, \textbf{8},12,16, \textbf{24}[/latex] Multiples of [latex]8: \textbf{8},16, \textbf{24}[/latex] The least common denominator is [latex]8[/latex] — the smallest multiple they have in common. Rewrite [latex]\dfrac{3}{4}[/latex] with a denominator of [latex]8[/latex]. You have to multiply both the top and bottom by [latex]2[/latex] so you don't change the relationship between them (because [latex]\dfrac{2}{2}=1[/latex]).
[latex]\dfrac{3}{4}\cdot\dfrac{2}{2}=\dfrac{6}{8}[/latex]
We don't need to rewrite [latex]\dfrac{5}{8}[/latex] since it already has the common denominator.Answer
Both [latex]\dfrac{6}{8}[/latex] and [latex]\dfrac{5}{8}[/latex] have the same denominator, and you can describe how much pizza is left with common terms, and compare the values more easily.Adding Fractions with Unlike Denominators
- Find a common denominator.
- Rewrite each fraction using the common denominator.
- Now that the fractions have a common denominator, you can add the numerators.
- 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 [latex]\dfrac{6}{9}[/latex] you can rewrite [latex]6[/latex] and [latex]9[/latex] using the smallest factors possible as follows:[latex]\dfrac{6}{9}=\dfrac{2\cdot3}{3\cdot3}[/latex]
Since there is a [latex]3[/latex] in both the numerator and denominator, and fractions can be considered division, we can divide the [latex]3[/latex] in the top by the [latex]3[/latex] in the bottom to reduce to [latex]1[/latex].[latex]\dfrac{6}{9}=\dfrac{2\cdot\cancel{3}}{3\cdot\cancel{3}}=\dfrac{2}{3}\cdot1=\dfrac{2}{3}[/latex]
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 [latex]\dfrac{2}{3}+\dfrac{1}{5}[/latex]. Simplify the answer.Answer: Since the denominators are not alike, find a common denominator by multiplying the denominators.
[latex]3\cdot5=15[/latex]
Rewrite each fraction with a denominator of [latex]15[/latex] by multiplying it by [latex]1[/latex].[latex]\begin{array}{c}\dfrac{2}{3}\cdot\dfrac{5}{5}=\dfrac{10}{15}\\\\\dfrac{1}{5}\cdot\dfrac{3}{3}=\dfrac{3}{15}\end{array}[/latex]
Add the fractions by adding the numerators and keeping the denominator the same. Make sure the fraction cannot be simplified.[latex]\dfrac{10}{15}+\dfrac{3}{15}=\dfrac{13}{15}[/latex]
Answer
[latex-display]\dfrac{2}{3}+\dfrac{1}{5}=\dfrac{13}{15}[/latex-display]Example
Add [latex]\dfrac{3}{7}+\dfrac{2}{21}[/latex]. 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 [latex]7: 7, 14, \textbf{21}[/latex] Multiples of [latex]21:\textbf{21}[/latex] Rewrite each fraction with a denominator of [latex]21[/latex].
[latex]\begin{array}{c}\dfrac{3}{7}\cdot\dfrac{3}{3}=\dfrac{9}{21}\\\\\dfrac{2}{21}\end{array}[/latex]
Add the fractions by adding the numerators and keeping the denominator the same. Make sure the fraction cannot be simplified.[latex]\dfrac{9}{21}+\dfrac{2}{21}=\dfrac{11}{21}[/latex]
Answer
[latex-display]\dfrac{3}{7}+\dfrac{2}{21}=\dfrac{11}{21}[/latex-display]Think About It
Add [latex]\dfrac{3}{4}+\dfrac{1}{6}+\dfrac{5}{8}[/latex]. 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 [latex]4, 6[/latex], and [latex]8[/latex].
[latex]4=2\cdot2\\6=3\cdot2\\8=2\cdot2\cdot2\\\text{LCM}:\,\,2\cdot2\cdot2\cdot3=24[/latex]
Rewrite each fraction with a denominator of [latex]24[/latex].[latex]\begin{array}{c}\dfrac{3}{4}\cdot\dfrac{6}{6}=\dfrac{18}{24}\\\\\dfrac{1}{6}\cdot\dfrac{4}{4}=\dfrac{4}{24}\\\\\dfrac{5}{8}\cdot\dfrac{3}{3}=\dfrac{15}{24}\end{array}[/latex]
Add the fractions by adding the numerators and keeping the denominator the same.[latex]\dfrac{18}{24}+\dfrac{4}{24}+\dfrac{15}{24}=\dfrac{37}{24}[/latex]
Write the improper fraction as a mixed number and simplify the fraction.[latex]\dfrac{37}{24}=\normalsize 1\,\,\dfrac{13}{24}[/latex]
Answer
[latex-display]\dfrac{3}{4}+\dfrac{1}{6}+\dfrac{5}{8}=\normalsize 1\dfrac{13}{24}[/latex-display]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 [latex]\dfrac{1}{5}-\dfrac{1}{6}[/latex]. 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.
[latex]5\cdot6=30[/latex]
Rewrite each fraction as an equivalent fraction with a denominator of [latex]30[/latex].[latex]\begin{array}{c}\dfrac{1}{5}\cdot\dfrac{6}{6}=\dfrac{6}{30}\\\\\dfrac{1}{6}\cdot\dfrac{5}{5}=\dfrac{5}{30}\end{array}[/latex]
Subtract the numerators. Simplify the answer if needed.[latex]\dfrac{6}{30}-\dfrac{5}{30}=\dfrac{1}{30}[/latex]
Answer
[latex-display]\dfrac{1}{5}-\dfrac{1}{6}=\dfrac{1}{30}[/latex-display]Example
Subtract [latex]\dfrac{5}{6}-\dfrac{1}{4}[/latex]. Simplify the answer.Answer: Find the least common multiple of the denominators—this is the least common denominator. Multiples of [latex]6: 6, \textbf{12}, 18, 24[/latex] Multiples of [latex]4: 4, 8, \textbf{12},16, 20[/latex] [latex]12[/latex] is the least common multiple of [latex]6[/latex] and [latex]4[/latex]. Rewrite each fraction with a denominator of [latex]12[/latex].
[latex]\begin{array}{c}\dfrac{5}{6}\cdot\dfrac{2}{2}=\dfrac{10}{12}\\\\\dfrac{1}{4}\cdot\dfrac{3}{3}=\dfrac{3}{12}\end{array}[/latex]
Subtract the fractions. Simplify the answer if needed.[latex]\dfrac{10}{12}-\dfrac{3}{12}=\dfrac{7}{12}[/latex]
Answer
[latex-display]\dfrac{5}{6}-\dfrac{1}{4}=\dfrac{7}{12}[/latex-display]Licenses & Attributions
CC licensed content, Original
- Revision and Adaptiation. Provided by: Lumen Learning License: CC BY: Attribution.
CC licensed content, Shared previously
- Ex: Add Fractions with Unlike Denominators (Basic with Model). Authored by: Mathispower4u. License: CC BY: Attribution.
- Ex: Subtract Fractions with Unlike Denominators (Basic with Model) Mathispower4u . Authored by: Mathispower4u. License: CC BY: Attribution.
- Unit 2: Fractions and Mixed Numbers, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology Located at: https://www.nroc.org/. License: CC BY: Attribution.
