Arithmetic With 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
- Multiply two or more fractions
- Multiply a fraction by a whole number
- Find the reciprocal of a number
- Divide a fraction by a whole number
- Divide a fraction by a fraction
Introduction
One of the most common things people forget from arithmetic is how to add and subtract fractions. This section will remind you how to do this. You will also get to practice both adding and subtracting fractions as well as multiplying and dividing them. As you work through the rest of this course, remember to come back to this section for a quick reminder of these skills if the process becomes fuzzy. 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 3 and 2 are factors of 6
- numerator: the top part of a fraction - the numerator in the fraction [latex]\frac{2}{3}[/latex] is 2
- denominator: the bottom part of a fraction - the denominator in the fraction [latex]\frac{2}{3}[/latex] is 3
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 | Perform the indicated mathematical operations including addition, subtraction, multiplication, division |
| 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 4 pieces, and someone takes 1 piece. Now, [latex]\frac{1}{4}[/latex] of the pizza is gone and [latex]\frac{3}{4}[/latex] remains. Note that both of these fractions have a denominator of 4, 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 8 equal parts and 3 of those parts were gone, leaving [latex]\frac{5}{8}[/latex]?
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, 4, etc. For example, find the least common multiple of 2 and 5.
| First, list all the multiples of 2: | 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 using common terms.Answer: Rewrite the fractions [latex] \frac{3}{4}[/latex] and [latex]\frac{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 4: 4, 8, 12, 16 Multiples of 8: 8, 16, 24 The least common denominator is 8—the smallest multiple they have in common. Rewrite [latex] \frac{3}{4}[/latex] with a denominator of 8. You have to multiply both the top and bottom by 2 so you don't change the relationship between them.
[latex] \frac{3}{4}\cdot \frac{2}{2}=\frac{6}{8}[/latex]
We don't need to rewrite [latex] \frac{5}{8}[/latex] since it already has the common denominator.Answer
Both [latex]\frac{6}{8}[/latex] and [latex] \frac{5}{8}[/latex] have the same denominator, and you can describe how much pizza is left with common terms.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]\frac{6}{9}[/latex] you can rewrite 6 and 9 using the smallest factors possible as follows:[latex]\frac{6}{9}=\frac{2\cdot3}{3\cdot3}[/latex]
Since there is a 3 in both the numerator and denominator, and fractions can be considered division, we can divide the 3 in the top by the 3 in the bottom to reduce to 1.[latex]\frac{6}{9}=\frac{2\cdot\cancel{3}}{3\cdot\cancel{3}}=\frac{2\cdot1}{3}=\frac{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] \frac{2}{3}+\frac{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 15.[latex]\begin{array}{c}\frac{2}{3}\cdot \frac{5}{5}=\frac{10}{15}\\\\\frac{1}{5}\cdot \frac{3}{3}=\frac{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] \frac{10}{15}+\frac{3}{15}=\frac{13}{15}[/latex]
Answer
[latex-display] \frac{2}{3}+\frac{1}{5}=\frac{13}{15}[/latex-display]Example
Add [latex] \frac{3}{7}+\frac{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 7: 7, 14, 21 Multiples of 21: 21 Rewrite each fraction with a denominator of 21.
[latex]\begin{array}{c}\frac{3}{7}\cdot \frac{3}{3}=\frac{9}{21}\\\\\frac{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] \frac{9}{21}+\frac{2}{21}=\frac{11}{21}[/latex]
Answer
[latex-display] \frac{3}{7}+\frac{2}{21}=\frac{11}{21}[/latex-display]Think About It
Add [latex] \frac{3}{4}+\frac{1}{6}+\frac{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 4, 6, and 8.
[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 24.[latex]\begin{array}{c}\frac{3}{4}\cdot \frac{6}{6}=\frac{18}{24}\\\\\frac{1}{6}\cdot \frac{4}{4}=\frac{4}{24}\\\\\frac{5}{8}\cdot \frac{3}{3}=\frac{15}{24}\end{array}[/latex]
Add the fractions by adding the numerators and keeping the denominator the same.[latex]\frac{18}{24}+\frac{4}{24}+\frac{15}{24}=\frac{37}{24}[/latex]
Write the improper fraction as a mixed number and simplify the fraction.[latex] \frac{37}{24}=1\,\,\frac{13}{24}[/latex]
Answer
[latex-display]\frac{3}{4}+\frac{1}{6}+\frac{5}{8}=1\frac{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]\frac{1}{5}-\frac{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 30.[latex]\begin{array}{c}\frac{1}{5}\cdot \frac{6}{6}=\frac{6}{30}\\\\\frac{1}{6}\cdot \frac{5}{5}=\frac{5}{30}\end{array}[/latex]
Subtract the numerators. Simplify the answer if needed.[latex] \frac{6}{30}-\frac{5}{30}=\frac{1}{30}[/latex]
Answer
[latex-display] \frac{1}{5}-\frac{1}{6}=\frac{1}{30}[/latex-display]Example
Subtract [latex]\frac{5}{6}-\frac{1}{4}[/latex]. Simplify the answer.Answer: Find the least common multiple of the denominators—this is the least common denominator. Multiples of 6: 6, 12, 18, 24 Multiples of 4: 4, 8 12, 16, 20 12 is the least common multiple of 6 and 4. Rewrite each fraction with a denominator of 12.
[latex]\begin{array}{c}\frac{5}{6}\cdot \frac{2}{2}=\frac{10}{12}\\\\\frac{1}{4}\cdot \frac{3}{3}=\frac{3}{12}\end{array}[/latex]
Subtract the fractions. Simplify the answer if needed.[latex]\frac{10}{12}-\frac{3}{12}=\frac{7}{12}[/latex]
Answer
[latex-display] \frac{5}{6}-\frac{1}{4}=\frac{7}{12}[/latex-display]
By dividing each fourth in half, you can divide the candy bar into eighths.
Then, choose half of those to get [latex]\frac{3}{8}[/latex].
In both of the above cases, to find the answer, you can multiply the numerators together and the denominators together.
Multiplying Two Fractions
[latex-display] \large\frac{a}{b}\cdot \frac{c}{d}=\frac{a\cdot c}{b\cdot d}=\frac{\text{product of the numerators}}{\text{product of the denominators}}[/latex-display]Multiplying More Than Two Fractions
[latex-display] \large\frac{a}{b}\cdot \frac{c}{d}\cdot \frac{e}{f}=\frac{a\cdot c\cdot e}{b\cdot d\cdot f}[/latex-display]Example
Multiply [latex] \large\frac{2}{3}\cdot \frac{4}{5}[/latex].Answer: Multiply the numerators and multiply the denominators.
[latex]\large \frac{2\cdot 4}{3\cdot 5}[/latex]
Simplify, if possible. This fraction is already in lowest terms.[latex] \large\frac{8}{15}[/latex]
Answer
[latex-display]\large\frac{8}{15}[/latex-display]- Given [latex] \frac{8}{15}[/latex], the factors of 8 are: 1, 2, 4, 8 and the factors of 15 are: 1, 3, 5, 15. [latex] \frac{8}{15}[/latex] is simplified because there are no common factors of 8 and 15.
- Given [latex] \frac{10}{15}[/latex], the factors of 10 are: 1, 2, 5, 10 and the factors of 15 are: 1, 3, 5, 15. [latex] \frac{10}{15}[/latex] is not simplified because 5 is a common factor of 10 and 15.
Think About It
Multiply [latex] \large\frac{2}{3}\cdot \frac{1}{4}\cdot\frac{3}{5}[/latex]. Simplify the answer. What makes this example different than the previous ones? Use the box below to write down a few thoughts about how you would multiply three fractions together. [practice-area rows="2"][/practice-area]Answer: Multiply the numerators and multiply the denominators.
[latex]\large \frac{2\cdot 1\cdot 3}{3\cdot 4\cdot 5}[/latex]
Simplify first by canceling (dividing) the common factors of 3 and 2. 3 divided by 3 is 1, and 2 divided by 2 is 1.[latex]\begin{array}{c}\frac{2\cdot 1\cdot3}{3\cdot (2\cdot 2)\cdot 5}\\\frac{\cancel{2}\cdot 1\cdot\cancel{3}}{\cancel{3}\cdot (\cancel{2}\cdot 2)\cdot 5}\\\frac{1}{10}\end{array}[/latex]
Answer
[latex-display]\large\frac{2}{3}\cdot \frac{1}{4}\cdot\frac{3}{5}[/latex] = [latex]\large\frac{1}{10}[/latex-display]Divide Fractions
There are times when you need to use division to solve a problem. For example, if painting one coat of paint on the walls of a room requires 3 quarts of paint and you have a bucket that contains 6 quarts of paint, how many coats of paint can you paint on the walls? You divide 6 by 3 for an answer of 2 coats. There will also be times when you need to divide by a fraction. Suppose painting a closet with one coat only required [latex] \frac{1}{2}[/latex] quart of paint. How many coats could be painted with the 6 quarts of paint? To find the answer, you need to divide 6 by the fraction, [latex] \frac{1}{2}[/latex]. Before we begin dividing fractions, let's cover some important terminology.- reciprocal: two fractions are reciprocals if their product is 1 (Don't worry; we will show you examples of what this means.)
- quotient: the result of division
| Original number | Reciprocal | Product |
|---|---|---|
| [latex]\large\frac{3}{4}[/latex] | [latex]\large\frac{4}{3}[/latex] | [latex]\large\frac{3}{4}\cdot \frac{4}{3}=\frac{3\cdot 4}{4\cdot 3}=\frac{12}{12}=1[/latex] |
| [latex]\large\frac{1}{2}[/latex] | [latex]\large\frac{2}{1}[/latex] | [latex]\large\frac{1}{2}\cdot\frac{2}{1}=\frac{1\cdot}{2\cdot1}=\frac{2}{2}=1[/latex] |
| [latex] 3=\large\frac{3}{1}[/latex] | [latex]\large\frac{1}{3}[/latex] | [latex]\large\frac{3}{1}\cdot \frac{1}{3}=\frac{3\cdot 1}{1\cdot 3}=\frac{3}{3}=1[/latex] |
| [latex]2\large\frac{1}{3}=\large\frac{7}{3}[/latex] | [latex]\large\frac{3}{7}[/latex] | [latex]\large\frac{7}{3}\cdot\frac{3}{7}=\frac{7\cdot3}{3\cdot7}=\frac{21}{21}=1[/latex] |
Division by Zero
You know what it means to divide by 2 or divide by 10, but what does it mean to divide a quantity by 0? Is this even possible? Can you divide 0 by a number? Consider the fraction[latex]\large\frac{0}{8}[/latex]
We can read it as, “zero divided by eight.” Since multiplication is the inverse of division, we could rewrite this as a multiplication problem.[latex]\text{?}\cdot{8}=0[/latex].
We can infer that the unknown must be 0 since that is the only number that will give a result of 0 when it is multiplied by 8.
Now let’s consider the reciprocal of [latex]\frac{0}{8}[/latex] which would be [latex]\frac{8}{0}[/latex]. If we rewrite this as a multiplication problem, we will have[latex]\text{?}\cdot{0}=8[/latex].
This doesn't make any sense. There are no numbers that you can multiply by zero to get a result of 8. The reciprocal of [latex]\frac{8}{0}[/latex] is undefined, and in fact, all division by zero is undefined.
Caution! Division by zero is undefined and so is the reciprocal of any fraction that has a zero in the numerator. For any real number a, [latex]\large\frac{a}{0}[/latex] is undefined. Additionally, the reciprocal of [latex]\large\frac{0}{a}[/latex] will always be undefined.Divide a Fraction by a Whole Number
When you divide by a whole number, you are multiplying by the reciprocal. In the painting example where you need 3 quarts of paint for a coat and have 6 quarts of paint, you can find the total number of coats that can be painted by dividing 6 by 3, [latex]6\div3=2[/latex]. You can also multiply 6 by the reciprocal of 3, which is [latex] \frac{1}{3}[/latex], so the multiplication problem becomes[latex]\large\frac{6}{1}\cdot \frac{1}{3}=\frac{6}{3}=2[/latex].
Dividing is Multiplying by the Reciprocal
For all division, you can turn the operation into multiplication by using the reciprocal. Dividing is the same as multiplying by the reciprocal.[latex]\large\frac{1}{5}\text{ of }\frac{3}{4}=\frac{1}{5}\cdot \frac{3}{4}=\frac{3}{20}[/latex]
Each person gets [latex]\large\frac{3}{20}[/latex] of a whole candy bar.
If you have a recipe that needs to be divided in half, you can divide each ingredient by 2, or you can multiply each ingredient by [latex]\frac{1}{2}[/latex] to find the new amount. For example, dividing by 6 is the same as multiplying by the reciprocal of 6, which is [latex]\frac{1}{6}[/latex]. Look at the diagram of two pizzas below. How can you divide what is left (the red shaded region) among 6 people fairly?
Each person gets one piece, so each person gets [latex]\arge\frac{1}{4}[/latex] of a pizza.
Dividing a fraction by a whole number is the same as multiplying by the reciprocal, so you can always use multiplication of fractions to solve division problems.
Example
Find [latex]\large\frac{2}{3}\div 4[/latex].Answer: Write your answer in lowest terms. Dividing by 4 or [latex]\large\frac{4}{1}[/latex] is the same as multiplying by the reciprocal of 4, which is [latex] \frac{1}{4}[/latex].
[latex]\large\frac{2}{3}\div 4=\frac{2}{3}\cdot \frac{1}{4}[/latex]
Multiply numerators and multiply denominators.[latex]\large\frac{2\cdot 1}{3\cdot 4}=\frac{2}{12}[/latex]
Simplify to lowest terms by dividing numerator and denominator by the common factor 4.[latex]\large\frac{1}{6}[/latex]
Answer
[latex-display]\large\frac{2}{3}\div4=\frac{1}{6}[/latex-display]Example
Divide. [latex] 9\div\large\frac{1}{2}[/latex].Answer: Write your answer in lowest terms. Dividing by [latex]\large\frac{1}{2}[/latex] is the same as multiplying by the reciprocal of [latex]\large\frac{1}{2}[/latex], which is [latex]\large\frac{2}{1}[/latex].
[latex]9\div\large\frac{1}{2}=\frac{9}{1}\cdot\frac{2}{1}[/latex]
Multiply numerators and multiply denominators.[latex] \large\frac{9\cdot 2}{1\cdot 1}=\frac{18}{1}=18[/latex]
This answer is already simplified to lowest terms.Answer
[latex-display]9\div\large\frac{1}{2}=18[/latex-display]Divide a Fraction by a Fraction
Sometimes you need to solve a problem that requires dividing by a fraction. Suppose you have a pizza that is already cut into 4 slices. How many [latex]\large\frac{1}{2}[/latex] slices are there? |
 |
You would have 12 slices, which is the same as multiplying 4 by 3.
Dividing with Fractions
- Find the reciprocal of the number that follows the division symbol.
- Multiply the first number (the one before the division symbol) by the reciprocal of the second number (the one after the division symbol).
Example
Divide [latex]\large\frac{2}{3}\div \frac{1}{6}[/latex].Answer: Multiply by the reciprocal. KEEP [latex]\large\frac{2}{3}[/latex] CHANGE [latex] \div [/latex] to [latex]\cdot[/latex] FLIP [latex]\large\frac{1}{6}[/latex]
[latex]\large\frac{2}{3}\cdot \frac{6}{1}[/latex]
Multiply numerators and multiply denominators.[latex]\large\frac{2\cdot6}{3\cdot1}=\frac{12}{3}[/latex]
Simplify.[latex]\large\frac{12}{3}=4[/latex]
Answer
[latex-display]\large\frac{2}{3}\div \frac{1}{6}=4[/latex-display]Example
Divide [latex]\large\frac{3}{5}\div \frac{2}{3}[/latex].Answer: Multiply by the reciprocal. Keep [latex]\large\frac{3}{5}[/latex], change [latex] \div [/latex] to [latex]\cdot[/latex], and flip [latex] \frac{2}{3}[/latex].
[latex]\large\frac{3}{5}\cdot \frac{3}{2}[/latex]
Multiply numerators and multiply denominators.[latex]\large\frac{3\cdot 3}{5\cdot 2}=\frac{9}{10}[/latex]
Answer
[latex-display]\large\frac{3}{5}\div \frac{2}{3}=\frac{9}{10}[/latex-display]Licenses & Attributions
CC licensed content, Original
- Learning Activities. Authored by: Lumen Learning. License: CC BY: Attribution.
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
CC licensed content, Shared previously
- Ex 1: Divide Fractions (Basic). Authored by: James Sousa (Mathispower4u.com) . License: CC BY: Attribution.
- College Algebra. Authored by: Abramson, Jay et al.. Located at: https://openstax.org/books/prealgebra/pages/4-2-multiply-and-divide-fractions. License: CC BY: Attribution.
- Ex 1: Multiply Fractions (Basic) . Authored by: James Sousa (Mathispower4u.com) . License: CC BY: Attribution.
- Multiply and Divide Fractions. Provided by: OpenStax Located at: https://openstax.org/books/prealgebra/pages/4-2-multiply-and-divide-fractions. License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].
- Unit 2: Fractions and Mixed Numbers, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology and Education Located at: https://www.nroc.org/. License: CC BY: Attribution.
