5.3: Addition and Subtraction of Mixed Numbers

Fundamentals of Mathematics · LibreTexts

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    From LibreTexts, under CC BY, as the source states it; its equations are written in TeX, which the PDF shows typeset. This page at LibreTexts.

    Learning Objectives
    • be able to add and subtract mixed numbers
    Sample Set A

    Find the following sums and differences.

    \(8 \dfrac{3}{5} + 5 \dfrac{1}{4}\). Convert each mixed number to an improper fraction.

    Solution

    \(8 \dfrac{3}{5} = \dfrac{5 \cdot 8 + 3}{5} = \dfrac{40 + 3}{5} = \dfrac{43}{5}\)

    \(5 \dfrac{1}{4} = \dfrac{4 \cdot 5 + 1}{4} = \dfrac{20 + 1}{4} = \dfrac{21}{4}\). Now add the improper fractions \(\dfrac{43}{5}\) and \(\dfrac{21}{4}\).

    \(\dfrac{43}{5} + \dfrac{21}{4}\) The LCD = 20.

    \(\begin{array} {rcll} {\dfrac{43}{5} + \dfrac{21}{4}} & = & {\dfrac{43 \cdot 4}{20} + \dfrac{21 \cdot 5}{20}} & {} \\ {} & = & {\dfrac{172}{20} + \dfrac{105}{20}} & {} \\ {} & = & {\dfrac{172 + 105}{20}} & {} \\ {} & = & {\dfrac{277}{20}} & {\text{Convert this improper fraction to a mixed number.}} \\ {} & = & {13 \dfrac{17}{20}} & {} \end{array}\)

    Thus, \(8 \dfrac{3}{5} + 5 \dfrac{1}{4} = 13 \dfrac{17}{20}.\)

    Sample Set A

    \(3 \dfrac{1}{8} - \dfrac{5}{6}\). Convert the mixed number to an improper fraction.

    Solution

    \(3 \dfrac{1}{8} = \dfrac{3 \cdot 8 + 1}{8} = \dfrac{24 + 1}{8} = \dfrac{25}{8}\)

    \(\dfrac{25}{8} - \dfrac{5}{6}\) The LCD = 24.

    \(\begin{array} {rcll} {\dfrac{25}{8} - \dfrac{5}{6}} & = & {\dfrac{25 \cdot 3}{24} - \dfrac{5 \cdot 4}{24}} & {} \\ {} & = & {\dfrac{75}{24} - \dfrac{20}{24}} & {} \\ {} & = & {\dfrac{75 - 20}{24}} & {} \\ {} & = & {\dfrac{55}{24}} & {\text{Convert this improper fraction to a mixed number.}} \\ {} & = & {2 \dfrac{7}{24}} & {} \end{array}\)

    Thus, \(3 \dfrac{1}{8} - \dfrac{5}{6} = 2 \dfrac{7}{24}.\)

    Practice Set A

    Find the following sums and differences.

    \(1 \dfrac{5}{9} + 3 \dfrac{2}{9}\)

    Answer

    \(4 \dfrac{7}{9}\)

    Practice Set A

    \(10 \dfrac{3}{4} - 2 \dfrac{1}{2}\)

    Answer

    \(8 \dfrac{1}{4}\)

    Practice Set A

    \(2 \dfrac{7}{8} + 5 \dfrac{1}{4}\)

    Answer

    \(8 \dfrac{1}{8}\)

    Practice Set A

    \(8 \dfrac{3}{5} - \dfrac{3}{10}\)

    Answer

    \(8 \dfrac{3}{10}\)

    Practice Set A

    \(16 + 2 \dfrac{9}{16}\)

    Answer

    \(18 \dfrac{9}{16}\)

    Exercises

    For the following problems, perform each indicated opera­tion.

    Exercise \(\PageIndex{1}\)

    \(3 \dfrac{1}{8} + 4 \dfrac{3}{8}\)

    Answer

    \(7 \dfrac{1}{2}\)

    Exercise \(\PageIndex{2}\)

    \(5 \dfrac{1}{3} + 6 \dfrac{1}{3}\)

    Exercise \(\PageIndex{3}\)

    \(10 \dfrac{5}{12} + 2 \dfrac{1}{12}\)

    Answer

    \(12 \dfrac{1}{2}\)

    Exercise \(\PageIndex{4}\)

    \(15 \dfrac{1}{5} - 11 \dfrac{3}{5}\)

    Exercise \(\PageIndex{5}\)

    \(9 \dfrac{3}{11} + 12 \dfrac{3}{11}\)

    Answer

    \(21 \dfrac{6}{11}\)

    Exercise \(\PageIndex{6}\)

    \(1 \dfrac{1}{6} + 3 \dfrac{2}{6} + 8 \dfrac{1}{6}\)

    Exercise \(\PageIndex{7}\)

    \(5 \dfrac{3}{8} + 1 \dfrac{1}{8} - 2 \dfrac{5}{8}\)

    Answer

    \(3 \dfrac{7}{8}\)

    Exercise \(\PageIndex{8}\)

    \(\dfrac{3}{5} + 5 \dfrac{1}{5}\)

    Exercise \(\PageIndex{9}\)

    \(2 \dfrac{2}{9} - \dfrac{5}{9}\)

    Answer

    \(1 \dfrac{2}{3}\)

    Exercise \(\PageIndex{10}\)

    \(6 + 11 \dfrac{2}{3}\)

    Exercise \(\PageIndex{11}\)

    \(17 - 8 \dfrac{3}{14}\)

    Answer

    \(8 \dfrac{11}{14}\)

    Exercise \(\PageIndex{12}\)

    \(5 \dfrac{1}{3} + 2 \dfrac{1}{4}\)

    Exercise \(\PageIndex{13}\)

    \(6 \dfrac{2}{7} - 1 \dfrac{1}{3}\)

    Answer

    \(4 \dfrac{20}{21}\)

    Exercise \(\PageIndex{14}\)

    \(8 \dfrac{2}{5} + 4 \dfrac{1}{10}\)

    Exercise \(\PageIndex{15}\)

    \(1 \dfrac{1}{3} + 12 \dfrac{3}{8}\)

    Answer

    \(13 \dfrac{17}{24}\)

    Exercise \(\PageIndex{16}\)

    \(3 \dfrac{1}{4} + 1 \dfrac{1}{3} - 2 \dfrac{1}{2}\)

    Exercise \(\PageIndex{17}\)

    \(4 \dfrac{3}{4} - 3 \dfrac{5}{6} + 1 \dfrac{2}{3}\)

    Answer

    \(2 \dfrac{7}{12}\)

    Exercise \(\PageIndex{18}\)

    \(3 \dfrac{1}{12} + 4 \dfrac{1}{3} + 1 \dfrac{1}{4}\)

    Exercise \(\PageIndex{19}\)

    \(5 \dfrac{1}{15} + 8 \dfrac{3}{10} - 5 \dfrac{4}{5}\)

    Answer

    \(7 \dfrac{17}{30}\)

    Exercise \(\PageIndex{20}\)

    \(7 \dfrac{1}{3} + 8 \dfrac{5}{6} - 2 \dfrac{1}{4}\)

    Exercise \(\PageIndex{21}\)

    \(19 \dfrac{20}{21} + 42 \dfrac{6}{7} - \dfrac{5}{14} + 12 \dfrac{1}{7}\)

    Answer

    \(74 \dfrac{25}{42}\)

    Exercise \(\PageIndex{22}\)

    \(\dfrac{1}{16} + 4 \dfrac{3}{4} + 10 \dfrac{3}{8} - 9\)

    Exercise \(\PageIndex{23}\)

    \(11 - \dfrac{2}{9} + 10 \dfrac{1}{3} - \dfrac{2}{3} - 5 \dfrac{1}{6} + 6 \dfrac{1}{18}\)

    Answer

    \(21 \dfrac{1}{3}\)

    Exercise \(\PageIndex{24}\)

    \(\dfrac{5}{2} + 2 \dfrac{1}{6} + 11 \dfrac{1}{3} - \dfrac{11}{6}\)

    Exercise \(\PageIndex{25}\)

    \(1 \dfrac{1}{8} + \dfrac{9}{4} - \dfrac{1}{16} - \dfrac{1}{32} + \dfrac{19}{8}\)

    Answer

    \(5 \dfrac{21}{32}\)

    Exercise \(\PageIndex{26}\)

    \(22 \dfrac{3}{8} - 16 \dfrac{1}{7}\)

    Exercise \(\PageIndex{27}\)

    \(15 \dfrac{4}{9} + 4 \dfrac{9}{16}\)

    Answer

    \(20 \dfrac{1}{144}\)

    Exercise \(\PageIndex{28}\)

    \(4 \dfrac{17}{88} + 5 \dfrac{9}{110}\)

    Exercise \(\PageIndex{29}\)

    \(6 \dfrac{11}{12} + \dfrac{2}{3}\)

    Answer

    \(7 \dfrac{7}{12}\)

    Exercise \(\PageIndex{30}\)

    \(8 \dfrac{9}{16} - \dfrac{7}{9}\)

    Exercise \(\PageIndex{31}\)

    \(5 \dfrac{2}{11} - \dfrac{1}{12}\)

    Answer

    \(5 \dfrac{13}{132}\)

    Exercise \(\PageIndex{32}\)

    \(18 \dfrac{15}{16} - \dfrac{33}{34}\)

    Exercise \(\PageIndex{33}\)

    \(1 \dfrac{89}{112} - \dfrac{21}{56}\)

    Answer

    \(1 \dfrac{47}{212}\)

    Exercise \(\PageIndex{34}\)

    \(11 \dfrac{11}{24} - 7 \dfrac{13}{18}\)

    Exercise \(\PageIndex{35}\)

    \(5 \dfrac{27}{84} - 3 \dfrac{5}{42} + 1 \dfrac{1}{21}\)

    Answer

    \(3 \dfrac{1}{4}\)

    Exercise \(\PageIndex{36}\)

    \(16 \dfrac{1}{48} - 16 \dfrac{1}{96} + \dfrac{1}{144}\)

    Exercise \(\PageIndex{37}\)

    A man pours \(2 \dfrac{5}{8}\) gallons of paint from a bucket into a tray. After he finishes pouring, there are \(1 \dfrac{1}{4}\) gallons of paint left in his bucket. How much paint did the man pour into the tray?

    Hint:

    Think about the wording.

    Answer

    \(2 \dfrac{5}{8}\) gallons

    Exercise \(\PageIndex{38}\)

    A particular computer stock opened at \(37 \dfrac{3}{8}\) and closed at \(38 \dfrac{1}{4}\). What was the net gain for this stock?

    Exercise \(\PageIndex{39}\)

    A particular diet program claims that \(4 \dfrac{3}{16}\) pounds can be lost the first month, \(3 \dfrac{1}{4}\) pounds can be lost the second month, and \(1 \dfrac{1}{2}\) pounds can be lost the third month. How many pounds does this diet program claim a person can lose over a 3-month period?

    Answer

    \(8 \dfrac{15}{16}\) pounds

    Exercise \(\PageIndex{40}\)

    If a person who weighs \(145 \dfrac{3}{4}\) pounds goes on the diet program described in the problem above, how much would he weigh at the end of 3 months?

    Exercise \(\PageIndex{41}\)

    If the diet program described in the problem above makes the additional claim that from the fourth month on, a person will lose \(1 \dfrac{1}{8}\) pounds a month, how much will a person who begins the program weighing \(208 \dfrac{3}{4}\) pounds weight after 8 months?

    Answer

    \(194 \dfrac{3}{16}\) pounds

    Exercises for Review

    Exercise \(\PageIndex{42}\)

    Use exponents to write \(4 \cdot 4 \cdot 4\)

    Exercise \(\PageIndex{43}\)

    Find the greatest common factor of 14 and 20.

    Answer

    2

    Exercise \(\PageIndex{44}\)

    Convert \(\dfrac{16}{5}\) to a mixed number.

    Exercise \(\PageIndex{45}\)

    Find the sum. \(\dfrac{4}{9} + \dfrac{1}{9} + \dfrac{2}{9}\).

    Answer

    \(\dfrac{7}{9}\)

    Exercise \(\PageIndex{46}\)

    Find the difference. \(\dfrac{15}{26} - \dfrac{3}{10}\)

    This page titled 5.3: Addition and Subtraction of Mixed Numbers was last modified on Sat, 08 Jun 2024 18:05:14 GMT and is shared under a CC BY license and was authored, remixed, and/or curated by Denny Burzynski and Wade Ellis, Jr. (OpenStax CNX) via source content that was edited to the style and standards of the LibreTexts platform.