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13.6: Appendix- Working with Fractions

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    But I hate math...

    You knew all the arithmetic you will need for this course by the end of ninth grade, but it’s easy to get rusty. Don’t worry if you are, but do review the following material. If you have a little “math anxiety,” keep in mind that the key is to approach things slowly. Each of the basic concepts is relatively easy, and if you work to understand each point before going on to the next, you will be able to master the material. In fact, the only algebra you will need is the very minimal amount required to add and multiply rather simple fractions.

    Try to work through it in small steps, rather than trying to grasp everything all at one. As with much else in this course, you will also need to work through problems on your own. The most important factor in mastering any skill is practice.

    How Fractions Work

    A fraction consists of a numerator and a denominator. The numerator is the number on the top and the denominator is the number on the bottom. So, the numerator of 5/7 is 5, and the denominator is 7. Two fractions have a common denominator just in case they have the same denominator; 5/7 and 3/7 have a common denominator (namely 7), but 5/7 and 8/11 do not. It is often easier to work with fractions if we convert them into their decimal equivalents. To find the decimal equivalent for a fraction, divide the numerator by the denominator. For example, to convert 1/4 to a decimal, divide 1 by 4 (to get .25). To convert 3/5 to a decimal, divide 3 by 5 (to get .6). Such conversions are easy if you use a calculator (which you are encouraged to do).

    Adding Fractions

    To add two fractions that have a common denominator, you simply add their numerators and write it above their denominator. For example, 3/7 + 2/7 = 5/7. And 4/52 + 3/52 = 7/52.

    If you want to add fractions that have different denominators, you must find a common denominator. Once you do this, you simply add their numerators and write the result above the common denominator. In many cases, finding a common denominator is straightforward, but you can avoid such worries if you replace the fractions by their decimal equivalents and simply add those.

    Example: Add 3/5 + 1/4. You can either find a common denominator or you can add their decimal equivalents.

    Common Denominator

    The lowest common denominator of 3/5 and 1/4 is 20. So, we can express 3/5 as 12/20, and 1/4 as 5/20. And 12/20 + 4/20 = 17/20.

    Decimal Equivalents

    The decimal equivalent of 3/5 is .6 (divide 3 by 6 to get this) and the decimal equivalent of 1/4 is .25 (divide 1 by 4). So, 3/5 + 2/3 = .6 + .25 + .85

    We can check our two approaches by seeing whether they yield the same result; is .85 equal to 17/20? To answer this, we divide 17 by 20, which is .85, just as it should be.

    Multiplying Fractions

    To multiply fractions, you just multiply their numerators to get the new numerator and you multiply their denominators to get the new denominator.

    Example: What is 3/5 x 3/4? Multiply the two numerators (3 x 3) to get the new numerator, which is 9, then multiply the two denominators (5 x 4) to get the new denominator, which is 20. Putting these together, the answer is 9/20.

    Example: What is 4/52 x 3/51? Multiply the numerators to get 12, and the denominators to get 2652. So, the answer is 12/2652 (which reduces to 1/222).

    You can also always convert fractions to their decimal equivalents and then multiply them. We won’t worry much at the beginning about reducing fractions.

    But do note that when you multiply fractions you must multiply their denominators as well as their numerators. 4/52 x 3/52 is not 12/52 (it’s 12/(52 x 52)). Probabilities range from zero to one, and most of our calculations will involve fractions between zero and one. There are two very important points to remember about such fractions.

    1. When you add one such fraction to another, the result will be larger than either fraction alone.
    2. When you multiply one such fraction by another, the result will be smaller than either fraction alone.

    Exercises

    Find the value of each of the following:

    1. 2/3 + 1/3
    2. 2/6 + 1/6
    3. 2/3 + 1/6 (you need a common denominator here)
    4. 4/9 + 11/20
    5. 2/3 x 1/3 (the denominator here will be 9, not 3.)
    6. 2/6 x 1/6
    7. 2/3 x 1/6 (when we multiply fractions we don’t use a common denominator).
    8. 2/6 x 1/3
    9. 4/9 x 11/20
    10. 4/52 x 1/51

    This page titled 13.6: Appendix- Working with Fractions is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Jason Southworth & Chris Swoyer via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.