# Dividing Fractions

## Overview

This workbook was created through the efforts of three instructors at Scottsdale Community College in Scottsdale, Arizona. Any individual may download and utilize a digital copy of this workbook for free.

# Dividing Fractions

Before we discuss division with fractions, let’s take a step back and talk just about division in general. What does the operation of division do? Take a look at the problem below.

Problem | Explanation | Result | Check |

## \(6 \div 3\) | There are exactly 2 pieces of size 3 inside 6. | ## \(6\div3=2\) | ## \(2*3=6\) |

What did we do? We multiplied 6 by the reciprocal of 3. Thas is, we multiplied 6 by \(\frac{1}{3}\) and we achieved the same result as \(6\div3\). Division, then, can be converted to multiplication by using the reciprocal.

Let's see if this process make logical sense when we divide by a fraction.

Suppose you want to share a candy bar with 3 friends. You kno each of your friends would get \(\frac{1}{3}\) of the bar. You want to be sure so you ask. "How many pieces of size \(\frac{1}{3}\) are there in one candy bar?"

Multiplication | Multiplication Steps | Result | Check |

## \(1\div\frac{1}{3}\) | ## \(1\div\frac{1}{3}=1*3=3\) | ## \(1\div\frac{1}{3}=3\) | ## \(3*\frac{1}{3}=\frac{3}{3}=1\) |

We changed divising by \(\frac{1}{3}\) to multiplication by 3 (The reciprocal of \(\frac{1}{3}\)) giving us a result of 3. There are indeed 3 pieces of size \(\frac{1}{3}\) inside one full candy bar. Our division process works for fractions as well.

Let’s look at one more example just to be sure you have the idea.

You and some friends sit down to dinner and discover there are \(3\frac{1}{2}\) rolls left from your meal the night before. You decide to split all the rolls into \(\frac{1}{2}\) size pieces (except for the one that already is) and then divvy them up between you. How many \(\frac{1}{2}\) size pieces will there be?

We start with 3 full-size rolls and half of a roll as seen below.

Then, we break the rolls in half and cound the total number of \(\frac{1}{2}\) size pieces.

From the diagram, we can see easily that there are 7 pieces of size \(\frac{1}{2}\). So, depending on how many friends you have eating with you that night, you can pass out the pieces and maybe keep some extra for yourself. :-0)

What would the mathematics look like for this problem?

Problem | Multiplication Steps | Result | Check |

## \(3\frac{1}{2}\div\frac{1}{2}\) | \(3\frac{1}{2}\div\frac{1}{2}=\frac{7}{2}\div\frac{1}{2}\space \text{Convert}\space 3\frac{1}{2}\text{to an improper fraction} \) \(=\frac{7}{2}*\frac{2}{1}\space\text{Multiply by reciprocal of}\space \frac{1}{2}.\) \(=\frac{7}{1} \space\text{Remove the common factor 2.}\) \(= 7\space \text{Simplify to get final result.}\) | ## \(3\frac{1}{2}\div\frac{1}{2}=7\) | ## \(7*\frac{1}{2}=\frac{7}{1}*\frac{1}{2}\)## \(=\frac{7}{2}=3\frac{1}{2}\) |

**Steps to divide fractions (full list):**

- Convert any whole numbers to fractions (over 1).
- Convert any mixed numbers to improper fractions.
- Change DIVISION to MULTIPLICATION TIMES THE RECIPROCAL of the SECOND fraction.
- Multiply straight across.
- Reduce along the way if possible (only after switching to multiplication).
- Present final, reduced answer at the end.

**NOTE: We do not need to obtain a common denominator when dividing fractions!**

**Example 5: **Divide each of the following. If applicable, wright your answer as *both* an improper fration *and* a mixed number.

# a. \(\frac{2}{3}\div11 =\)

# b. \(2\div\frac{2}{5}=\)

# c. \(\frac{7}{2}\div\frac{3}{4}=\)

# d. \(\frac{8}{12}\div4=\)

# e. \(3\frac{1}{2}\div5\frac{3}{8}=\)

# f. \(3\div2=\)

**You Try**

**Example 6: **Divide each of the following. If applicable, wright your answer as *both* and improper fraction *and* a mixed number.

# a. \(\frac{2}{3} \div11=\)

# b. \(\frac{1}{5}\div7=\)

# c. \(3\frac{1}{4}\div\frac{1}{2}=\)

# Application of Fraction Division

**Example 7:** If part of a recipe for Albondigas Soup calls for 3 small potatoes, \(1\frac{1}{2}\) cups of salsa and 2 pounds of ground beef, how much of each of these ingredients would be needed to make half of the recipe?

**GIVEN:**

**GOAL:**

**MATH WORK:**

**CHECK:**

**FINAL RESULT AS A COMPLETE SENTENCE:**

**You Try**

**Example 8: **Sally was cutting a large tree into log sections that would fit into her fireplace. If her fireplace would take a log that was \(1\frac{1}{4}\) feet long and her tree was 100 feet long, how many sections of \(1\frac{1}{4}\) feet length would she cut out of the tree?

**GIVEN:**

**GOAL:**

**MATH WORK:**

**CHECK:**

**FINAL RESULT AS A COMPLETE SENTENCE:**

# Exponents/Order of Operations

Remember again our order of operations from Lesson 1? We will use the same order when working with fraction expressions that involve multiple operations and exponents.

P | Simplify items inside Parenthesis ( ), brackets [ ] or other grouping symbols first. |

E | Simplify items that are raised to powers (Exponents) |

MD | Perform Multiplication and Division next (as they appear from Left to Right) |

AS | Perform Addition and Subtraction on what is left. (as they appear from Left to Right) |

**Example 9: **Evaluate. If applicable, wright your answer as *both* an improper fraction *and *a mixed number.

# a. \((\frac{3}{4})^2=\)

# b. \(\frac{3}{5}(\frac{2}{3})^2=\)

**Example 10: **Evaluate: If applicable, write your answer as *both* an improper fraction *and* a mixed number.

# a. \(\frac{3}{4}+\frac{4}{5}\div\frac{2}{3}=\)

# b. \((2-\frac{1}{3})^2\space \div\space (\frac{1}{4}+\frac{1}{6})=\)

**You Try**

**Example 11: **Evaluate. If applicable, wright your answer as *both* an improper fraction* and* a mixed number.

# a. \((\frac{3}{7})^3=\)

# b. \(\frac{4}{5}\div(\frac{2}{3})^2=\)

**Special Cases**

**Example 12: **What happens when you multiply by 0?

# \((\frac{3}{4} - \frac{1}{3})*(\frac{2}{4}-\frac{1}{2})\)

**Example 13: **What happens when you divide by 0?

# \(\frac{2}{3}\div\frac{0}{1}\)

**You Try**

**Example 14:** Bill earns $10 for every hour he works each week up to 40 hours. Any additional hours are considered overtime and he earns "time and a half" wages. If he worked 56 hours one week, what were his total earnings?

**GIVEN:**

**GOAL:**

**MATH WORK:**

**CHECK:**

**FINAL RESULT AS A COMPLETE SENTENCE:**