5.3: Addition and Subtraction of Mixed Numbers
- be able to add and subtract mixed numbers
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, \(8 \dfrac{3}{5} + 5 \dfrac{1}{4} = 13 \dfrac{17}{20}.\)
\(3 \dfrac{1}{8} - \dfrac{5}{6}\). 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?
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. \(\dfrac{15}{26} - \dfrac{3}{10}\)