6.9: Combinations of Operations with Decimals and Fractions

Learning Objectives

• be able to combine operations with decimals

Having considered operations with decimals and fractions, we now consider opera­tions that involve both decimals and fractions.

Sample Set A

Perform the following operations.

\(0.38 \cdot \dfrac{1}{4}\). Convert both numbers to decimals or both numbers to fractions. We’ll convert to decimals.

Solution

\(\begin{array} {r} {.25} \\ {4 \overline{)1.00}} \\ {\underline{\ \ 8\ \ }} \\ {20} \\ {\underline{20}} \\ {0} \end{array}\)

To convert \(\dfrac{1}{4}\) to a decimal, divide 1 by 4.

Now multiply 0.38 and .25.

\(\begin{array} {r} {^1\ \ \ } \\ {^4 \ \ \ } \\ {.38} \\ {\underline{\times .25}} \\ {190} \\ {\underline{76\ \ }} \\ {.0950} \end{array}\)

Thus, \(0.38 \cdot \dfrac{1}{4} = 0.095\).

In the problems that follow, the conversions from fraction to decimal, or decimal to fraction, and some of the additions, subtraction, multiplications, and divisions will be left to you.

Sample Set A

\(1.85 + \dfrac{3}{8} \cdot 4.1\). Convert \(\dfrac{3}{8}\) to a decimal.

Solution

\(1.85 + 0.375 \cdot 4.1\) Multiply before adding.

\(1.85 + 1.5375\) Now add.

3.3875

Sample Set A

\(\dfrac{5}{13} (\dfrac{4}{5} - 0.28)\) Convert 0.28 to a fraction.

Solution

\(\begin{array} {rcl} {\dfrac{5}{13} (\dfrac{4}{5} - \dfrac{28}{100}} & = & {\dfrac{5}{13} (\dfrac{4}{5} - \dfrac{7}{25})} \\ {} & = & {\dfrac{5}{13} (\dfrac{20}{25} - \dfrac{7}{25})} \\ {} & = & {\dfrac{\begin{array} {c} {^1} \\ {\cancel{5}} \end{array}}{\begin{array} {c} {\cancel{13}} \\ {^1} \end{array}} \cdot \dfrac{\begin{array} {c} {^1} \\ {\cancel{13}} \end{array}}{\begin{array} {c} {\cancel{25}} \\ {^5} \end{array}}} \\ {} & = & {\dfrac{1}{5}} \end{array}\)

Sample Set A

\(\begin{array} {rcll} {\dfrac{0.125}{1\dfrac{1}{3}} + \dfrac{1}{16} - 0.1211} & = & {\dfrac{\dfrac{125}{1000}}{\dfrac{4}{3}} + \dfrac{1}{16} - 0.1211} & {} \\ {} & = & {\dfrac{\dfrac{1}{8}}{\dfrac{4}{3}} + \dfrac{1}{16} - 0.1211} & {} \\ {} & = & {\dfrac{1}{8} \cdot \dfrac{3}{4} + \dfrac{1}{16} - 0.1211} & {} \\ {} & = & {\dfrac{3}{32} + \dfrac{1}{16} - 0.1211} & {} \\ {} & = & {\dfrac{3}{32} + \dfrac{1}{16} - 0.1211 = \dfrac{5}{32} - 0.1211} & {} \\ {} & = & {0.15625 - 0.1211} & {} \\ {} & = & {0.03515} & {\text{ Convert this to fraction form}} \\ {} & = & {\dfrac{3515}{100,000}} & {} \\ {} & = & {\dfrac{703}{20,000}} & {} \end{array}\)

Practice Set A

Perform the following operations.

\(\dfrac{3}{5} + 1.6\)

Answer

2.2 or \(2 \dfrac{1}{5}\)

Practice Set A

\(8.91 + \dfrac{1}{5} \cdot 1.6\)

Answer

9.23

Practice Set A

\(1 \dfrac{9}{16} (6.12 + \dfrac{7}{25})\)

Answer

10

Practice Set A

\(\dfrac{0.156}{1 \dfrac{11}{15}} - 0.05\)

Answer

\(\dfrac{1}{25}\) or 0.04

Exercises

Exercise \(\PageIndex{1}\)

\(\dfrac{3}{10} + 0.7\)

Answer

1

Exercise \(\PageIndex{2}\)

\(\dfrac{1}{5} + 0.1\)

Exercise \(\PageIndex{3}\)

\(\dfrac{5}{8} - 0.513\)

Answer

0.112

Exercise \(\PageIndex{4}\)

\(0.418 - \dfrac{67}{200}\)

Exercise \(\PageIndex{5}\)

\(0.22 \cdot \dfrac{1}{4}\)

Answer

0.055

Exercise \(\PageIndex{6}\)

\(\dfrac{3}{5} \cdot 8.4\)

Exercise \(\PageIndex{7}\)

\(\dfrac{1}{25} \cdot 3.19\)

Answer

0.1276

Exercise \(\PageIndex{8}\)

\(\dfrac{3}{20} \div 0.05\)

Exercise \(\PageIndex{9}\)

\(\dfrac{7}{40} \div 0.25\)

Answer

0.7

Exercise \(\PageIndex{10}\)

\(1 \dfrac{1}{15} \div 0.9 \cdot 0.12\)

Exercise \(\PageIndex{11}\)

\(9.26 + \dfrac{1}{4} \cdot 0.81\)

Answer

9.4625

Exercise \(\PageIndex{12}\)

\(0.588 + \dfrac{1}{40} \cdot 0.24\)

Exercise \(\PageIndex{13}\)

\(\dfrac{1}{20} + 3.62 \cdot \dfrac{3}{8}\)

Answer

1.4075

Exercise \(\PageIndex{14}\)

\(7 + 0.15 \div \dfrac{3}{30}\)

Exercise \(\PageIndex{15}\)

\(\dfrac{15}{16} \cdot (\dfrac{7}{10} - 0.5)\)

Answer

0.1875

Exercise \(\PageIndex{16}\)

\(0.2 \cdot (\dfrac{7}{20} + 1.1143)\)

Exercise \(\PageIndex{17}\)

\(\dfrac{3}{4} \cdot (0.875 + \dfrac{1}{8})\)

Answer

0.75

Exercise \(\PageIndex{18}\)

\(5.198 - 0.26 \cdot (\dfrac{14}{250} + 0.119)\)

Exercise \(\PageIndex{19}\)

\(0.5 \dfrac{1}{4} + (0.3)^2\)

Answer

0.615

Exercise \(\PageIndex{20}\)

\((1.4)^2 - 1.6 \dfrac{1}{2}\)

Exercise \(\PageIndex{21}\)

\((\dfrac{3}{8})^2 - 0.000625 + (1.1)^2\)

Answer

1.35

Exercise \(\PageIndex{22}\)

\((0.6)^2 \cdot (\dfrac{1}{20} - \dfrac{1}{25})\)

Exercise \(\PageIndex{23}\)

\((\dfrac{1}{2})^2 - 0.125\)

Answer

0.125

Exercise \(\PageIndex{24}\)

\(\dfrac{0.75}{4 \dfrac{1}{2}} + \dfrac{5}{12}\)

Exercise \(\PageIndex{25}\)

\((\dfrac{0.375}{2 \dfrac{1}{16}} - \dfrac{1}{33})\)

Answer

\(0.\overline{15}\)

Exercise \(\PageIndex{26}\)

\(8 \dfrac{1}{3} \cdot (\dfrac{1 \dfrac{1}{4}}{2.25} + \dfrac{9}{25})\)

Exercise \(\PageIndex{27}\)

\(\dfrac{\dfrac{0.32}{\dfrac{12}{35}}}{0.35}\)

Answer

\(2.\overline{6}\)

Exercise \(\PageIndex{28}\)

\(\dfrac{(\sqrt{\dfrac{49}{64}} - 5)0.125}{1.375}\)

Exercises for Review

Exercise \(\PageIndex{29}\)

Is 21,480 divisible by 3?

Answer

yes

Exercise \(\PageIndex{30}\)

Expand \(14^4\). Do not find the actual value.

Exercise \(\PageIndex{31}\)

Find the prime factorization of 15,400.

Answer

\(2^3 \cdot 5^2 \cdot 7 \cdot 11\)

Exercise \(\PageIndex{32}\)

Convert 8.016 to a fraction.

Exercise \(\PageIndex{33}\)

Find the quotient.

Answer

\(0.\overline{592}\)