5.3: Addition and Subtraction of Mixed Numbers

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