• Unit 4: Fractions and Rational Numbers

    Fractions are amazingly helpful; we can use them to tell so many different kinds of stories, but this can also make them seem overwhelming or confusing. How likely will a coin flip result in heads? We use a fraction to express the answer to this probability \frac{1}{2}: our outcome (a heads up) should occur 1 out of 2 times or one-half of the time.

    If you serve 10 pizzas to 40 students, how much pizza should each student receive? We use fractions to determine that each student should receive \frac{1}{4} or one-fourth of a single pizza. We also use fractions to discuss ratios and rates, such as this car gets 40 miles per gallon.

    Combining fractions using addition, subtraction, multiplication, or division is a bit more complicated than combining integers, but these combinations are useful in many situations and problems.

    For example, if your friend Andy has one-half of a chocolate bar and Tai has one-third of a bar, how much do they have together? To answer this question, we need to add the rational numbers \frac{1}{2}+ \frac{1}{3} (see Section 4.8). In this unit, we explore these kinds of topics, beginning with a basic review of how to notate a fraction and ending with a discussion on how to use them in multi-step problems.

    Completing this unit should take you approximately 8 hours.

    • Upon successful completion of this unit, you will be able to:

      • identify parts of a fraction;
      • recognize fractions in lowest terms;
      • recognize equivalent fractions;
      • identify improper fractions and mixed numbers;
      • convert between mixed numbers and improper fractions;
      • determine the least common denominator between fractions;
      • use equivalent fractions to add and subtract fractions and mixed numbers with like and unlike denominators;
      • solve multiplication and division problems with fractions and mixed numbers; and
      • solve real-world math problems involving fractions.

    • 4.1: Identifying Parts of Fractions

      A first step toward understanding fractions is to review their different parts – like we identified the exponent and base in an exponential expression in section 3.4.

      For example, the fraction \frac{8}{11}  uses three symbols: two integers (eight and 11) and a horizontal line called the fraction bar. We call the number on top of the bar the numerator and the number beneath the bar the denominator. You can read our example in several ways: 8 over 11, 8 out of 11, 8 parts per 11, or 8 divided by 11.
      • Fractions of Whole Numbers Book

        Read this section to learn more about the history and notation of fractions, how to visualize them in relation to integers, and how to identify the different parts (numerator and denominator). Complete the practice questions and check your answers.

      • Visualize Fractions Book

        Read this section which offers some useful ways to visualize fractions to help you understand the ideas they can represent. Complete the practice questions and check your answers.

    • 4.2: Equivalent Fractions

      Just as different words can represent the same concept (the synonyms big and large), different mathematical expressions can represent the same quantity. For example, imagine a pizza cut into four slices. Each slice represents \frac{1}{4} of the pizza. If you eat two of the four slices, you can say you ate \frac{2}{4} of the entire pizza. As this image indicates, our two \frac{1}{4} slices make up an entire \frac{1}{2} of the pizza.



      This is all to say that for our pizza example, \frac{2}{4}=\frac{1}{2}. Each of these fractions has a different numerator and denominator, but they both represent the same quantity or value. We say these two fractions are equivalent. Equivalent fractions may be written to look different, but they represent the same value.

      To practice, can you determine equivalent ways to write the fraction \frac{2}{6}? Are \frac{3}{4} and \frac{75}{100} equivalent? How can you tell?
      • Demonstrating Equivalent Fractions Page

        Watch this video for practice with equivalent fractions. It uses number lines to show how different fractions are equivalent, or the same value.

    • 4.3: Proper and Improper Fractions

      We call a fraction improper when the numerator is larger than its denominator. For example, \frac{10}{3}. There is nothing wrong or bad about improper fractions; they occur all around us, like regular fractions. Imagine, for example, sharing 10 candy bars among three people. Everyone will receive \frac{10}{3} of a candy bar.

    • 4.4: Mixed Numbers and Improper Fractions

      Let's say you just bought a full candy bar. Then you discover you already had a half-eaten candy bar in your pocket. How much do you have altogether? You might write that you have 1+\frac{1}{2} candy bars, which some math teachers would call a mixed number. It contains a regular whole number and a new fraction. In fact, some math teachers rewrite this expression without the addition sign altogether as:

      1+\frac{1}{2}=1\frac{1}{2}

      While this notation for mixed numbers is common in early math courses, mathematicians and math students largely abandon it once they move on to other courses, and they do so for good reason. It is very easy to confuse mixed number notation for multiplication, reading the above incorrectly as 1 times \frac{1}{2}. It is useful to keep in mind that mixed number notation hides a + sign, one that connects a whole number and a fraction. Nevertheless, it can be useful to understand and apply such notation.

      But it is more important for you to understand that every mixed number is simply another way of writing an improper fraction. In our example above, you can also think about the one whole candy bar as two halves, so that 1=\frac{1}{2}+\frac{1}{2}. This allows you to count how much candy you have in a different way. You can combine your two halves with the half candy bar in your pocket, so you have a total of three halves.

      In purely math terms, 1+\frac{1}{2}=1\frac{1}{2}=\frac{3}{2}, which is an improper fraction.
      • Visualize Fractions Book

        Indeed, we can convert every mixed number into an improper fraction. We will discuss this conversion trick in the next section. Read this text, complete the practice problems, and check your answers.

    • 4.5: Converting Between Improper Fractions and Mixed Numbers

      While it is rare for a math text to take a bold stance on almost anything, we are going to make an exception here! No one should ever use mixed number notation, especially in mathematical situations. This being said, we often see "mixed numbers" in our daily lives, most notably at gas stations. The image below shows a price of \displaystyle 327\frac{9}{10} pennies, which is, of course, a mixed number.

      Photo of a sign at a gas station listing the price for gas.

      Photo: Wikicommons

      One benefit of mixed number notation is it makes it easier to compare the quantity to whole numbers. For instance, we know that the 327\frac{9}{10} price is in between 327 and 328. When we convert this mixed number into an improper fraction, we obtain:

      327\frac{9}{10} = 327 + \frac{9}{10} = \frac{3270 + 9}{10} = \frac{3279}{10}.

      The second video link below describes how to convert mixed numbers into an improper fraction. Note that while many prefer to use proper and improper fractions to notate rational numbers, these expressions make it hard to determine which improper fraction lies between consecutive whole numbers. As much as we hate to admit it, this one point supports the use of mixed numbers.

      Of course, you can disfigure a lovely, beautiful, and clearly-expressed improper fraction to create an ugly, inferior mixed number – there is a downgrading conversion process in this direction, too. For example, consider the improper fraction 13/3 which we can rewrite as:

      \frac{13}{3} = \frac{12+1}{3} = \frac{12}{3} + \frac{1}{3} = 4+\frac{1}{3} = 4\frac{1}{3}

    • 4.6: Fractions in Lowest Terms

      Converting a mixed number into an improper fraction is an example of how to add fractions. Before we discuss how to add and subtract fractions more generally in section 4.8, we need to address a related issue that will help you add and subtract fractions more easily.

      Let's return to your aunt's coffee shop for a delicious and motivating example. Your aunt recently added a second breakfast sandwich to her menu, the Vegan Supreme. Like her original Bacon Breakfast Sandwich, her customers love her new addition.

      However, your aunt can only prepare one type of breakfast sandwich next weekend due to some scheduled kitchen maintenance. She decides to conduct a survey to help choose which one she should offer and learns that 600 out of 1,000 customers prefer the Vegan Supreme. We can record this figure as a fraction, namely \frac{600}{1000}, but it turns out that this fraction is equivalent to a smaller, easier-to-read-and-understand expression. Namely:

      \frac{600}{1000} = \frac{3}{5}

      It is easier to interpret this smaller-looking-but-equivalent fraction than to work with the original figure 600/1000. When we reduce the complex appearance of the original fraction – a process we call reducing the fraction or expressing it in lowest terms – it is clear that your aunt should offer the Vegan Supreme. Three-fifths of the customers she surveyed preferred it!

      Reducing fractions not only helps us make sandwich decisions. It also makes arithmetic computations easier. Reducing fractions to their lowest terms simplifies and clarifies how we combine these numbers via our standard operations.

    • 4.7: Finding Common Denominators

      Now that you are comfortable reducing fractions into their lowest terms, we want to build some familiarity with rewriting fractions. Specifically, we want to rewrite two fractions so that they have the same denominator.

      Consider the fractions 3/12 and 4/12. These two expressions have a common denominator (12). Now practice your fraction-reducing skills from the previous section to note that:

      \frac{3}{12} = \frac{1}{4} and \frac{4}{12} = \frac{1}{3}

      The fractions 1/3 and 1/4 can each be re-expressed so that they have a common denominator of 12. Something similar is also true for the fractions 20/60 and 24/60; they have a common denominator of 60, and each of these fractions reduces to 1/3 and 2/5.

      The process of finding a single common denominator for two fractions is, in some sense, the reverse of what we have just discussed: start with two different-denominator fractions and then rewrite them so they have the same denominator.

      The video you will watch next discusses how to find "the least common denominator". For our two previous examples, the least common denominator of 1/4 and 1/3 is 12, but the least common denominator of 1/3 and 2/5 is 15 (not 60, the larger denominator we used above). Indeed, instead of writing:

      \frac{1}{3} = \frac{20}{60} and \frac{2}{5} = \frac{24}{60},

      we could have written:

      \frac{1}{3} = \frac{5}{15} and \frac{2}{5} = \frac{6}{15}.

      To find the least common denominator, you will use the least common multiple of the denominators, a nice callback to section 2.2.
      • Finding Common Denominators Page

        Watch this video. Pause it after the narrator presents the second example to see if you can figure out the common denominator yourself. Then watch the rest of the video to see the solution.

      • Before we move on, we need to recognize that you can simply multiply the numbers to easily find a common denominator. But this method does not always produce the least common denominator.

        Consider the two fractions 3/8 and 5/6. You can compute the least common denominator by calculating the least common multiple LCM(8, 6), but if you only need a single common denominator, then you can simply multiply 8\times 6 = 48. Multiply the fraction with the eight denominator by sixes (one on top, one on bottom), and multiply the fraction with the six denominator by eights (one on top, one on bottom) as follows:

        \dfrac{3}{8} = \dfrac{6\times 3}{6\times 8} = \dfrac{18}{48} and \dfrac{5}{6} = \dfrac{8\times 5}{8\times 6} = \dfrac{40}{48}

        Now, we have common denominators! You can do this for any fractions a/b and c/d:

        \dfrac{a}{b} = \dfrac{d\times a}{d\times b} and \dfrac{c}{d} = \dfrac{b\times c}{b\times d}

    • 4.8: Adding and Subtracting Fractions

      When we add or subtract fractions, it is helpful to consider two different cases: when the fractions have the same denominator and when the fractions have different denominators. We first focus on adding and subtracting fractions that have the same or common denominator. It is a relatively short process, as we simply combine the numerators.

      For instance, consider adding 9/15 + 2/15. Because the denominators match, we are able to compute:

      \frac{9}{15} + \frac{2}{15} = \frac{9+2}{15} = \frac{11}{15}.

      Similarly:

      \frac{9}{15} - \frac{2}{15} = \frac{9-2}{15} = \frac{7}{15}.

      More generally, we have:

      \frac{a}{b} \pm \frac{c}{b} = \frac{a\pm c}{b}.
      • Add and Subtract Fractions with Common Denominators Book

        Read this text. Pay special attention to the sections on fraction addition and subtraction. They provide an overview of how to add and subtract fractions with the same denominator. Complete the practice questions and check your answers.

      • But what about adding and subtracting fractions that do not have the same denominator? This is where your work from section 4.7 comes to the rescue! First, rewrite the fractions so they do have the same denominators, and then add them like we did above!

        Here are some examples.

        Consider the sum \dfrac{1}{3} + \dfrac{1}{4}. Since the denominators are different, we cannot (yet) combine the numerators. Follow these steps!

        • First Step: Find a Common Denominator
          • Note that you can use any common denominator. This may be quicker, but you will probably have to reduce your answer at the end.
          • Using the least common denominator will keep the numbers somewhat small.

        • Second Step: Add or Subtract the Common Denominator Fractions

        • Possible Third Step: Reduce the final result in lowest terms
          For our example, we can use the least common denominator: LCM(3,4) = 12.

        • First Step. \dfrac{1}{3} = \dfrac{4}{12} and \dfrac{1}{4} = \dfrac{3}{12}

        • Second Step. \dfrac{4}{12} + \dfrac{3}{12} = \dfrac{4+3}{12} = \dfrac{7}{12}


        Here is another example. Let's try to subtract 3/8 from 5/6. Again, we see two different denominators. So our first step is to find a common one. Let's use the "easy" common denominator 8\times 6 = 48. We will find 3/8 = 18/48 and 5/6=40/48. Our next step is to subtract these expressions:

        \dfrac{40}{48} - \dfrac{18}{48} = \dfrac{40-18}{48} = \dfrac{22}{48}

        If we want to express this answer in lowest terms, we can write:

        \dfrac{22}{48} = \dfrac{11}{24}
      • Add and Subtract Fractions with Different Denominators Book

        Read this text for more examples and guidance. It gives a good review of how to find common denominators. Pay attention to the "How To" section at the beginning for an overview of the process. Do Examples 4.67 – 4.72 and check your answers.

      • More with Fractions Page

        Watch this video for more examples of how to add and subtract fractions with like and unlike denominators.

    • 4.9: Adding and Subtracting Mixed Numbers

      There are two different methods for adding and subtracting mixed numbers. One way is to treat the whole numbers and fractions separately. The other approach is to convert mixed numbers into improper fractions and then use the addition and subtraction methods we learned in the last two sections. As avid anti-mixed-number mathematicians, we, of course, prefer the latter, but both are worth exploring.

      • Add Mixed Numbers Book

        Read this section which explains the two methods for adding mixed numbers with common denominators. Do Examples 4.83 and 4.84 using the method of adding the whole numbers and fractions separately. Then, do Examples 4.85 using the method of converting mixed numbers to improper fractions. Be sure to check your answers.

      • Subtract Mixed Numbers Book

        Next, read this section which describes two methods for subtracting mixed numbers with common denominators. Pay attention to the gray "How To" boxes for an overview of the methods. Complete Example 4.90 using the whole numbers and fraction method, and Example 4.91 using the improper fraction method. Be sure to check your answers.

      • Practice with Unlike Denominators Book

        Complete this assessment to practice adding and subtracting unlike denominators. Be sure to check your answers.

    • 4.10: Applications of Adding and Subtracting Fractions

      Adding and subtracting fractions and mixed numbers comes up in a variety of real-world applications. Let's return to your aunt's coffee shop. Your aunt has placed some online ads to reach even more customers, and several of the ads include coupons. One fine Saturday morning, one of your regular patrons presents two coupons. One reads, "half off". The second reads, "one-third off".

      Assuming your aunt allows her customers to combine the coupons, how much of a reduction will they enjoy? Try to see if you can figure this out!

      We need to add \frac{1}{2} and \frac{1}{3}, which first requires a common denominator. We can write:

      \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}

      Our frugal customer only has to pay one-sixth of the original price; after you take off five-sixths!

      One last thought about our coupon-combining diner: if their total bill was $30.36, how much do they owe after they apply both coupons? The answer is in the next section, since this calculation requires multiplying fractions.

    • 4.11: Multiplying Fractions and Mixed Numbers

      Now we are ready to learn how to multiply fractions and mixed numbers.

    • 4.12: Dividing Fractions and Mixed Numbers

      Great news! Your aunt's coffee shop is so successful that she is opening up a second shop in a different neighborhood. While everyone is excited, you and your aunt are busy arranging the details for her grand opening. You and your aunt will be cutting a ribbon to commemorate the event, and you have already purchased 10 feet of ribbon to use. Only now, you realize you only need 1/2 feet of fabric for the event, and you wonder: how many 1/2 foot-long sections of fabric are in your 10-foot ribbon?

      This simple question amounts to trying to divide 10 by 1/2. You may be able to puzzle out the answer (there are 20 1/2-foot-long sections in your 10-foot-long ribbon!).
      • Dividing Fractions Page

        Watch these videos to figure out how to divide lots of numbers by fractions. As usual, pay careful attention to the worked examples.

      • Dividing Mixed Numbers Page

        Watch these videos to learn how to divide mixed numbers and see a few examples.

      • Dividing Fractions Exercises Book

        Complete this assessment to practice dividing fractions and check your answers. If you need more practice, do Problem Sets 2 and 3.

    • 4.13: Applications of Multiplying and Dividing Fractions

      Ribbon questions are not the only ones that fraction multiplication and division can answer! Let's look at some other examples you might run into.