Adding and Subtracting Fractions
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    To understand the sections below, it is necessary to know how to reduce fractions. It is strongly recommended that you first review how to reduce fractions before proceeding.

    esson: Reducing Fractions

    Reducing Fractions requires dividing the numerator and the denominator of a fraction by a common factor. Scaling up fractions is the reverse.

    First, let us look at a fundamental rule of mathematics.

    Consider this fraction.

fraction 2/3

    We could multiply this fraction by the number 1. Doing so would not change its value.

fraction 2/3 times 1

    Since 4/4 is also equal to 1, we can multiply 2/3 by 4/4 and not change its value.

fraction 2/3 scaled up to 12ths

    Reducing 8/12 by dividing the top and bottom of the fraction by 4, we getů

fraction 2/3

    So, we can scale fractions up however we please. Look at these fractions, each being multiplied by 1 to scale them up.

fractions scaled up table

    It can be seen that scaling up fractions is the opposite of Reducing Fractions.

    This skill will be used in the following sections.

    Look at this example.

adding two fractions

    Both fractions are referring to 11ths. Therefore, the numerators can be added while keeping the same denominators, like this.

adding two fractions

    Since the fraction cannot be reduced (the numerator and denominator have no common factor, other than 1), 7/11 is the final answer.

    The fractions with this next example can be combined with little effort.

adding two fractions

    The reason why this can be handled with little effort is because all the fractions have the same denominator. They are all in 8ths. To combine them, keep the same denominator and combine all of the numerators, like so.

adding two fractions

    Cleaning this up, we getů

6/8 fraction

    This fraction can be reduced.

reducing fraction

    So, 3/4 is the final solution.

    The next sections will explain how to add/subtract fractions that have unlike denominators.

    Assume you were asked to add these fractions.

adding fractions unlike denominators

    Notice that the fractions have unlike denominators. We cannot add them until the denominators are equal.

    First, we will have to decide what the common denominator can be. To get this common denominator, consider the multiples of the denominators, which are 9 and 6.

finding a common denominator multiple

    By looking at the table above, we can see that 9 and 6 have a common multiple. It is 18, which has been colored red.

    Next, we need to scale up the factors so that the new denominators have this common multiple. To help the process, the problem can be arranged vertically, like so.

adding fractions vertical

    We must change the denominators to that of their common multiple, which is 18. With that in mind, new fractions next to the existing fractions will be made but so that the new denominators are 18.

adding fractions vertical common denominator

    Since the 9 in the denominator must be multiplied by 2 to get 18, the 4 must also be multiplied by 2.

adding fractions vertical common denominator

    Likewise, the 6 in the other denominator must be multiplied by 3 to get 18; so, the numerator must be multiplied by 3, too.

adding fractions vertical common denominator

    The new fractions can be obtained by multiplying across, as the next graphic will indicate.

adding fractions vertical common denominator multiply

    Now that the denominators are equal, we can add the fractions on the right to get this answer.

adding fractions vertical different denominators

    Since 11 and 18 share no common factor (other than 1), it cannot be reduced, which makes it the final answer.

    ideo: Adding and Subtracting Fractions
    uiz: Adding and Subtracting Fractions

    The last section explains how to add fractions with a considerable amount of detail. This section will provide slightly less detail.

    Here is a subtraction problem.

subtracting fractions different denominators

    First, we must search for a common multiple using the denominators, 8, and 5.

getting common denominator

    Now that we have the common denominator of 40, we can set up our problem vertically. 40 has been placed in the fractions to the right.

subtracting fractions different denominators

    We need to multiply the first denominator by 5 and the second denominator by 8 and then multiply across.

subtracting fractions common denominator vertical

    Since the denominators are now equal, we can subtract the numerators.

subtracting fractions different denominators

    11 and 40 have no common factors (other than 1), which makes 11/40 the final solution.

    ideo: Adding and Subtracting Fractions
    uiz: Adding and Subtracting Fractions

    Use this instructional video to understand how to add and subtract fractions.

    ideo: Adding and Subtracting Fractions

    Use this interactive quiz to determine if you can add and subtract fractions.

    uiz: Adding and Subtracting Fractions

    Try these lessons, which are related to the sections above.

    esson: Reducing Fractions
    esson: Multiplying Fractions
    esson: Dividing Fractions
    esson: Adding Rational Expressions