In this module, we’ll provide a review of the basic mathematical operations (addition, subtraction, multiplication & division) with fractions. 

Module 2 Learning Objectives

At the completion of this module (and its respective sections), you will be able to:

  1. Examine the terminology of Fractions, such as Numerator and Denominator (CLO 1)

  2. Analyze the limitation of Fractions (CLO 6)

  3. Perform the operations of Fractions including Addition, Subtraction, Multiplication, and Division (CLO 1)

  4. Apply the operations of fractions to solve word problems. (CLO 1, 4, 5)

Fractions

Introduction to Fractions

Parts of a Whole

The concept of fractions is simply to break up something in equal parts. In other words, let say we take a sheet of paper and we cut it (as evenly as possible) into 10 different strips. Then we take 4 of those strips and give it to our kid. We can say our kid has 4 out of 10 strips, or \(\frac{4}{10}\) th’s of the sheet of paper.

You can also visualize it using pizza is cut into 8 slices. If you eat 2 slices of that pizza, then you can say that you’ve eaten \(\frac{2}{8}\) th’s of that pizza.

See a pattern? The number that represents the “whole” part is the bottom number and the number that represents what’s taken is the top number. In other words…

Figure 1: Breakdown of Fractions
Figure 1: Breakdown of Fractions
   

Definition - Numerator and Denominator

  • Numerator: The Top Number of a Fraction
    •  
  • Denominator: The Bottom Number of a Fraction

   

Definition - Proper vs Improper Fractions

  • Proper Fractions: When the numerator is smaller than the denominator
    •  
  • Improper Fractions: When the numerator is larger than the denominator

 

Definition - Zero in Fractions

Remember our lesson with Zero in Division? This works in the same fashion.

  • When Zero is the numerator: The answer is always zero
    •  
  • When Zero is the denominator: The answer is always undefined.

 

Definition - Reducing Fractions

When dealing with fractions, you always want to reduce it to lowest, or simplest terms. Check out the graphic above.

  • Look at the yellow 1 6 th row. If you add 3 of those 1 6 th tiles, you get 3 6 ths . Now, if you scroll up, 3 of those 1 6 th tiles are the same size as 1 of those blue 1 2 tiles. Meaning, 3 6 is the same as 1 2 .
    •  
  • Another, more common way to reduce fractions is determining the largest number that both the numerator and denominator can be divided by. In this case,the largest number that "goes into" 3 and 6 is 3. Thus, we divide the top and bottom numbers by three: 3 ÷ 3 6 ÷ 3 = 1 2

 

Example 1

Reduce the following to lowest terms:

\(\Large \frac{25}{55}\)

\[\begin{align*} \frac{25}{55}& \hspace{1.3cm}\mbox{Original Fraction}\\[1.5ex] \frac{25 \div 5}{55 \div 5}& \hspace{1.3cm}\mbox{Largest Number Goes into Both is 5}\\[1.5ex] \frac{5}{11}& \hspace{1.3cm}\mbox{Simplifed}\\[1.5ex] \end{align*}\]

Example 2

\(\Large \frac{42}{63}\)

\[\begin{align*} \frac{42}{63}& \hspace{1.3cm}\mbox{Original Fraction}\\[1.5ex] \frac{42 \div 7}{63 \div 7}& \hspace{1.3cm}\mbox{A Number Goes into Both is 7}\\[1.5ex] \frac{6}{9}& \hspace{1.3cm}\mbox{Simplifed}\\[1.5ex] \end{align*}\] WAIT! We can still reduce this fraction to lowest terms. The largest number that goes into both 6 and 9 is 3. Therefore, we can reduce it again: \[\begin{align*} \frac{6}{9}& \hspace{1.3cm}\mbox{Reduced Fraction from Above}\\[1.5ex] \frac{6 \div 3}{9 \div 3}& \hspace{1.3cm}\mbox{Largest Number Goes into Both is 3}\\[1.5ex] \frac{2}{3}& \hspace{1.3cm}\mbox{Simplifed}\\[1.5ex] \end{align*}\] I did it in this way because it may not always be clear to see the largest number that goes into both numbers. Therefore, you can see what number goes into both numbers, than you can keep reducing until you can’t do it anymore.

Example 3

\(\Large \frac{390}{260}\)

\[\begin{align*} \frac{390}{260}& \hspace{1.3cm}\mbox{Original Fraction}\\[1.5ex] \frac{390 \div 10}{260 \div 10}& \hspace{1.3cm}\mbox{Because the numbers end in zeros, 10 goes into both}\\[1.5ex] \frac{39}{26}& \hspace{1.3cm}\mbox{Simplifed-ish}\\[1.5ex] \frac{39\div 13}{26\div 13}& \hspace{1.3cm}\mbox{Largest number Goes into Both is 13}\\[1.5ex] \frac{3}{2}& \hspace{1.3cm}\mbox{Simplifed Completely}\\[1.5ex] \end{align*}\]

Practice Problems

Directions: It’s suggested to do these problems without a calculator. Reduce each of the fractions into lowest terms.

  1. \(\frac{9}{12}\)

  2. \(\frac{14}{36}\)

  3. \(\frac{10}{16}\)

  4. \(\frac{12}{16}\)

  5. \(\frac{7}{28}\)

  6. \(\frac{34}{51}\)

  7. \(\frac{8}{32}\)

  8. \(\frac{22}{55}\)

  9. \(\frac{85}{51}\)

  10. \(\frac{165}{121}\)

Multiplication and Division with Fractions

You’re probably wondering, why are we starting off with Multiplying and Dividing Fractions rather than Adding and Subtracting? The reason is because well… it’s so much easier! Trust me.

Definition - Multiplying Fractions

If you have two or more fractions, then to multiply them we have: a b × c d = a × c b × d In other words, when you multiply fractions, you multiply straight across.

 

Remark

A lot of students want to "criss-cross" the mulitplication. This is incorrect as this process deals with one fraction equaling another fraction, like you would see in proportions. Do not make this mistake.

   

Definition - Dividing Fractions

If you have two farctions, then to divide them we have: a b ÷ c d = a b × d c = a × d b × c In other words, to divide them, we will multiply the first fraction by the reciprocal or "flipped" second fraction.

 

Keep-Change-Flip

If you remember Keep-Change-Flip, this helps alot! This means, we keep the first fraction the same, change the sign to muliplication, flip the second fraction.

   

Example 1

Multiply the following fractions then reduce your answer to lowest terms.

\(\Large \frac{3}{16}\times \frac{1}{2}\)

\[\begin{align*} \frac{3}{16}\times \frac{1}{2}& \hspace{1.3cm}\mbox{Original Problem}\\[1.5ex] \frac{3 \times 1}{16\times 2}& \hspace{1.3cm}\mbox{Multiply Straight Across}\\[1.5ex] \frac{3}{32}& \hspace{1.3cm}\mbox{Simplifed, Final Answer}\\[1.5ex] \end{align*}\]

Example 2

\(\Large \frac{5}{8}\times \frac{2}{3}\)

\[\begin{align*} \frac{5}{8}\times \frac{2}{3}& \hspace{1.3cm}\mbox{Original Problem}\\[1.5ex] \frac{5 \times 2}{8\times 3}& \hspace{1.3cm}\mbox{Multiply Straight Across}\\[1.5ex] \frac{10}{24}& \hspace{1.3cm}\mbox{Simplified the Multiplication}\\[1.5ex] \frac{10\div 2}{24 \div 2}& \hspace{1.3cm}\mbox{Both Even Numbers, Divide by 2}\\[1.5ex] \frac{5}{12}& \hspace{1.3cm}\mbox{Simplified, Final Answer}\\[1.5ex] \end{align*}\]

Example 3

\(\Large \frac{1}{5}\times \frac{2}{7}\times \frac{3}{11}\)

\[\begin{align*} \frac{1}{5}\times \frac{2}{7}\times \frac{3}{11}& \hspace{1.3cm}\mbox{Original Problem}\\[1.5ex] \frac{1 \times 2 \times 3}{5\times 7 \times 11}& \hspace{1.3cm}\mbox{Multiply Straight Across}\\[1.5ex] \frac{6}{385}& \hspace{1.3cm}\mbox{Simplified the Multiplication, Final Answer}\\[1.5ex] \end{align*}\]

Example 4

\(\Large 8\times \frac{3}{4}\)

\[\begin{align*} 8\times \frac{3}{4}& \hspace{1.3cm}\mbox{Original Problem}\\[1.5ex] \frac{8}{1}\times \frac{3}{4}& \hspace{1.3cm}\mbox{Whole numbers can be written over 1}\\[1.5ex] \frac{8\times 3}{1\times 4}& \hspace{1.3cm}\mbox{Multiply Straight Across}\\[1.5ex] \frac{24}{4}& \hspace{1.3cm}\mbox{Simplified the Multiplication}\\[1.5ex] 6& \hspace{1.3cm}\mbox{Simplified, Final Answer}\\[1.5ex] \end{align*}\]

Example 5

\(\Large \frac{5}{16}\div \frac{15}{16}\)

\[\begin{align*} \frac{5}{16}\div \frac{15}{16}& \hspace{1.3cm}\mbox{Original Problem}\\[1.5ex] \frac{5}{16}\times \frac{16}{15}& \hspace{1.3cm}\mbox{Keep Change Flip}\\[1.5ex] \frac{5\times 16}{16\times 15}& \hspace{1.3cm}\mbox{Multiplied Straight Across}\\[1.5ex] \frac{5}{15}& \hspace{1.3cm}\mbox{16's Divide Out}\\[1.5ex] \frac{1}{3}& \hspace{1.3cm}\mbox{Both Reduced by 3, Final Answer}\\[1.5ex] \end{align*}\]

Reducing

When you reduce fractions, you can reduce vertically and diagonally, but never straight across.

   

Example 6

\(\Large \frac{20}{21}\div \frac{15}{42}\)

\[\begin{align*} \frac{20}{21}\div \frac{15}{42}& \hspace{1.3cm}\mbox{Original Problem}\\[1.5ex] \frac{20}{21}\times \frac{42}{15}& \hspace{1.3cm}\mbox{Keep Change Flip}\\[1.5ex] \frac{4}{3}\times \frac{6}{3}& \hspace{1.3cm}\mbox{20 & 15 Divide by 5, 21 & 42 Divide by 7}\\[1.5ex] \frac{4}{1}\times \frac{2}{3}& \hspace{1.3cm}\mbox{3 & 6 Divided by 3}\\[1.5ex] \frac{8}{3}& \hspace{1.3cm}\mbox{Final Answer}\\[1.5ex] \end{align*}\]

Example 7

\(\Large \frac{5}{6}\div \frac{13}{4}\)

\[\begin{align*} \frac{5}{6}\div \frac{13}{4}& \hspace{1.3cm}\mbox{Original Problem}\\[1.5ex] \frac{5}{6}\times \frac{4}{13}& \hspace{1.3cm}\mbox{Keep Change Flip}\\[1.5ex] \frac{5}{3}\times \frac{2}{13}& \hspace{1.3cm}\mbox{6 & 4 reduced by 2}\\[1.5ex] \frac{10}{39}& \hspace{1.3cm}\mbox{Final Answer}\\[1.5ex] \end{align*}\]

Practice Problems

Directions: It’s suggested to do these problems without a calculator. Perform the indicated operation for each problem. Also, reduce each of the fractions into lowest terms.

  1. \(\frac{18}{42} \times \frac{14}{75}\)

  2. \(\frac{13}{91}\times \frac{34}{65}\)

  3. \(\frac{10}{8} \times \frac{9}{5}\)

  4. \(\frac{30}{42} \times \frac{7}{100}\)

  5. \(\frac{10}{18} \times \frac{9}{5}\)

  6. \(\frac{45}{55}\times \frac{0}{32}\)

  7. \(\frac{21}{5}\div \frac{10}{3}\)

  8. \(\frac{5}{6}\div \frac{13}{4}\)

  9. \(\frac{5}{8}\div 2\)

  10. If you have $35 and you spend $7 for a sandwich and iced tea, what fraction of your money have you spent on food? What fraction do you still have?

  11. If you go on a bicycle trip of 75 miles in the mountains and 20% (or \(\frac{1}{5}\) ) of the trip is downhill, what fraction of the trip is not downhill? How many miles are not downhill?

Addition and Subtraction with Fractions

Adding and subtracting fractions is one of the hot topics on the struggle bus for students. There are a few things to remember anytime you add or subtract with fractions.

 

Least Common Denominator (LCD)

In order to add or subtract fractions, your bottom numbers (denominators) must be the same. Otherwise, it does not work.

   

To find the least common denominator, we have to go back to our elementary school days and bring back those times tables.

  • Look at the denominators. Make a times-table or a mulitple list for each denominator.
  • The first number they have in common is your least-common denominator.

 

Example 1

Find the Least-Common Denominator for the following fractions:

\(\Large \frac{1}{2}\) and \(\Large \frac{1}{9}\)

 

Look at the bottom numbers, 2 and 9. Start making a times-table list for each of them.

2: \(2 \times 1 = 2, \hspace{.5cm} 2\times 2 = 4, \hspace{.5cm} 2\times 3 = 6, \ldots\)

So the times table list for 2 would be something like this:

\[2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, \ldots\] For 9, we do the same process: 9: \(9 \times 1 = 9, \hspace{.5cm} 9\times 2 = 18, \hspace{.5cm} 9\times 3 = 27, \ldots\) . So the times table list for 9 would be something like this: \[9, 18, 27, 36, 45, 54, 63, 72, 81, 90, \ldots\] If you look at both lists, the first number they have in common is \(18\). Therefore, the least common denominator for \(\Large \frac{1}{2}\) and \(\Large \frac{1}{9}\) is \(18\).

Example 2

Find the Least-Common Denominator for the following fractions:

\(\Large \frac{7}{9}, \Large \frac{31}{36}, \frac{13}{18}\)

 

The times-table list for each of the denominators are listed below:

9: \(9, 18, 27, 36, 45, 54, 63, 72, 81, 90, \ldots\)

36: \(36, 72, 108, 144, 180, 216, 252, 288, 324, 360, \ldots\)

18: \(18, 36, 54, 72, 90, 108, \ldots\)

Looking at both lists, the first number that all three lists have in common is 36. Therefore, the least common denominator would be 36.

Example 3

Find the Least-Common Denominator for the following fractions:

\(\Large \frac{5}{8}, \frac{1}{10}, 6, \frac{3}{4}\)

 

Remark

Here, we have a number that is not a fraction (or so it appears). Anytime you have a whole number, you can always write it over 1.

 

Now, our problem looks like this:

\(\Large \frac{5}{8}, \frac{1}{10}, \frac{6}{1}, \frac{3}{4}\)

The times-table list for each of the denominators are listed below:

8: \(8, 16, 24, 32, 40, 48, 56, 64, 72, \ldots\)

10: \(10, 20, 30, 40, 50, 60, 70, \ldots\)

1: \(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, \ldots, 38, 39, 40, 41, 42, \ldots\)

4: \(4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, \ldots\)

 

Looking at both lists, the first number that all three lists have in common is 40. Therefore, the least common denominator would be 40.

Adding and Subtracting Fractions

Remember, we can’t add or subtract fractions until we can find a least-common denominator. That’s why that was so crucial.

Example 1

Add the following fractions:

\(\Large \frac{1}{2} + \frac{1}{3}\)

From using the processes from the Least-Common Denominator, the LCD for these fractions would be 6. Therefore, we need to rewrite the fractions so that they have a denominator of 6.

\[\begin{align*} \frac{1}{2} + \frac{1}{3}& \hspace{1.3cm}\mbox{Original Problem}\\[1.5ex] \frac{ }{6} + \frac{ }{6 }& \hspace{1.3cm}\mbox{Denominators are rewritten as 6}\\[1.5ex] \end{align*}\]

Here, we have to ask ourselves:

  1. What times 2 gives us 6? The answer is 3. Therefore, we multiply the top and bottom of the original left fraction by 3.

  2. What times 3 gives us 6? The answer is 2. Therefore, we multiply the top and bottom of the original right fraction by 2.

Now, we have our fractions re-written as:

\[\begin{align*} \frac{3}{6} + \frac{2}{6}& \hspace{1.3cm}\mbox{Rewritten Fractions}\\[1.5ex] \frac{5}{6}& \hspace{1.3cm}\mbox{Add the Top Numbers, Final Answer}\\[1.5ex] \end{align*}\]

Example 2

Add the following fractions:

\(\Large \frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}\)

 

From using the processes from the Least-Common Denominator, the LCD for these fractions would be 12. Therefore, we need to rewrite the fractions so that they have a denominator of 12.

\[\begin{align*} \frac{1}{2} + \frac{1}{3}+ \frac{1}{4}& \hspace{1.3cm}\mbox{Original Problem}\\[1.5ex] \frac{ }{12} + \frac{ }{12} + \frac{ }{12}& \hspace{1.3cm}\mbox{Denominators are rewritten as 12}\\[1.5ex] \frac{6}{12} + \frac{4}{12} + \frac{3}{12}& \hspace{1.3cm}\mbox{Rewritten Fractions}\\[1.5ex] \frac{13}{12}& \hspace{1.3cm}\mbox{Add the Top Numbers, Final Answer}\\[1.5ex] \end{align*}\]

Example 3

Subtract the following fractions

\(\Large \frac{2}{3} - \frac{5}{8}\)

 

Subtraction works the same way as addition. Yes, we still have to find a least common denominator, but at the end, we’ll subtract the top numbers.

From using the processes from the Least-Common Denominator, the LCD for these fractions would be 24. Therefore, we need to rewrite the fractions so that they have a denominator of 24.

\[\begin{align*} \frac{2}{3} - \frac{5}{8}& \hspace{1.3cm}\mbox{Original Problem}\\[1.5ex] \frac{ }{24} - \frac{ }{24}& \hspace{1.3cm}\mbox{Denominators are rewritten as 24}\\[1.5ex] \frac{16}{24} - \frac{15}{24}& \hspace{1.3cm}\mbox{Rewritten Fractions}\\[1.5ex] \frac{1}{24}& \hspace{1.3cm}\mbox{Subtract the Top Numbers, Final Answer}\\[1.5ex] \end{align*}\]

Practice Problems

Directions: It’s suggested to do these problems without a calculator.

  1. \(\Large \frac{3}{7}+\frac{2}{7}\)

  2. \(\Large \frac{1}{12}+\frac{5}{36}+\frac{11}{24}\)

  3. \(\Large \frac{5}{27}+\frac{5}{18}\)

  4. \(\Large 1-\frac{13}{20}\)

  5. \(\Large \frac{3}{4}-\frac{5}{12}\)

  6. Which is larger, \(\Large \frac{2}{3}\) or \(\Large \frac{4}{5}\) ? How much larger?

  7. Creed bought 3 bags of mung beans to eat at his desk. Each bag weighs \(\Large \frac{1}{4}\) pound. He gives one bag to Dwight. How many pounds does Creed have left?

  8. You have 3 recruiters for your funeral home because you want to recruit new college graduates to join your team. The recruiting team meets to have lunch when they in the home office on the same day.

    • Andrew takes 6 days to cover his recruiting area
    • Lisa takes 9 days to cover her recruiting area
    • Abby takes 12 days to cover hers.

How often do they have lunch together?

  1. Arrange the following fractions from smallest to largest. Then, find the difference between the largest and smallest numbers.

\[\Large \frac{7}{12}, \frac{5}{9}, \frac{11}{20}\]

  1. You’re building your budget for the upcoming fiscal year for your business. You anticipate that \(\frac{2}{3}\) ’s of your budget will be used for payroll and \(\frac{1}{4}\) will be used for utilities. What fraction of your budget do you have left?