2.5 Converting Between Fractions and Decimal Numbers and Combined Order of Operations

Converting Decimal Numbers to Fractions

It is possible to convert terminating decimal numbers (e.g., 0.275) and repeating, non-terminating decimal numbers (e.g., 0.333333…) to fractions. However, there is no exact equivalent fraction for non-repeating, non-terminating decimal numbers (e.g., 0.837508…).

Converting Terminating Decimal Numbers to Fractions

Any non-repeating, terminating decimal number can be converted to a fraction by following these steps:

  1. Count the number of decimal places in the decimal number.
  2. Move the decimal point by that many places to the right, making it a whole number, and divide this whole number by 10 raised to the power of the number of decimal places counted in Step 1; for example, if there were 2 decimal places in the original number, we divide by [latex]10^2 = 100[/latex].
  3. Simplify (or reduce) the fraction.

Example 2.5-a: Converting Terminating Decimal Numbers to Fractions

Convert the following decimal numbers to their fractional equivalents:

  1. 3.75
  2. 0.015

Solution

  1. Converting 3.75 to its fractional equivalent:

    [latex]3.75[/latex]

    3.75 contains two decimal places. Therefore, move the decimal point two places to the right and divide by [latex]10^2 = 100[/latex].

    [latex]\displaystyle{= \frac{3.75}{100} = \frac{375}{100}}[/latex]

    (This is the same as multiplying by 100 and dividing by 100: [latex]\displaystyle{\frac{3.75 \times 100}{100} = \frac{375}{100}}[/latex])

    [latex]\displaystyle{= \frac{375}{100} = \frac{375 \div 25}{100 \div 25}}[/latex]

    Dividing by 25 to simplify,

    [latex]\displaystyle{= \frac{15}{4} = 3\frac{3}{4}}[/latex]

    Therefore, 3.75 converted to its fractional equivalent is [latex]\displaystyle{\frac{15}{4}}[/latex] or [latex]\displaystyle{3\frac{3}{4}}[/latex].

  2. Converting 0.015 to its fractional equivalent:

    [latex]0.015[/latex]

    0.015 contains three decimal places. Therefore, move the decimal point three places to the right and divide by [latex]10^3 = 1,000[/latex].

    [latex]\displaystyle{= \frac{0.015}{1,000} = \frac{15}{1,000}}[/latex]

    (This is the same as multiplying by 1,000 and dividing by 1,000: [latex]\displaystyle{\frac{0.015 \times 1,000}{1,000} = \frac{15}{1,000}}[/latex])

    [latex]\displaystyle{= \frac{15}{1,000} = \frac{15 \div 5}{1,000 \div 5}}[/latex]

    Dividing by 5 to simplify,

    [latex]\displaystyle{= \frac{3}{200}}[/latex]

    Therefore, 0.015 converted to its fractional equivalent is [latex]\displaystyle{\frac{3}{200}}[/latex].

Converting Repeating Decimal Numbers to Fractions

Any repeating, non-terminating decimal number can be converted to a fraction by following the procedure illustrated in the following examples.

Example 2.5-b: Converting Repeating Decimal Numbers to Fractions

Convert 0.777777… to a fraction.

Solution

Let 0.777777… be equal to a fraction [latex]x[/latex].

Therefore,

a. [latex]x = 0.777777... = 0.\overline{7}[/latex]

Multiplying both sides by 10,

b. [latex]10x = 7.777777... = 7.\overline{7}[/latex]

We now have two repeating decimal numbers, each with the same decimal portion.

Subtracting a from b,

[latex]10x - x = 7.\overline{7} - 0.\overline{7}[/latex]

[latex]9x = 7[/latex]

Dividing both sides by 9,

[latex]\displaystyle{x = \frac{7}{9}}[/latex]

Therefore, [latex]\displaystyle{0.777777... = \frac{7}{9}}[/latex].

Example 2.5-c: Converting Repeating Decimal Numbers to Fractions

Convert 0.655555… to a fraction.

Solution

Let 0.655555… be equal to a fraction [latex]x[/latex].

Therefore,

a. [latex]x = 0.655555... = 0.\overline{65}[/latex]

Multiplying both sides by 10,

b. [latex]10x = 6.555555... = 6.\overline{5}[/latex]

The decimal portions of the two decimal numbers are different. Multiplying both sides by 10 again,

c. [latex]100x = 65.555555... = 65.\overline{5}[/latex]

We now have two repeating decimal numbers with the same decimal portion (b and c).

Subtracting b from c,

[latex]100x - 10x = 65.\overline{5} - 6.\overline{5}[/latex]

[latex]90x = 59[/latex]

Dividing both sides by 90,

[latex]\displaystyle{x = \frac{59}{90}}[/latex]

Therefore, [latex]\displaystyle{0.655555... = \frac{59}{90}}[/latex].

Example 2.5-d: Converting Repeating Decimal Numbers to Fractions

Convert 0.353535… to a fraction.

Solution

Let 0.353535… be equal to a fraction [latex]x[/latex].

Therefore,

a. [latex]x = 0.353535... = 0.\overline{35}[/latex]

Multiplying both sides by 10,

b. [latex]10x = 3.535353... = 3.\overline{53}[/latex]

The decimal portions of the two decimal numbers are different. Multiplying both sides by 10 again,

c. [latex]100x = 35.353535... = 35.\overline{35}[/latex]

We now have two repeating decimal numbers with the same decimal portion (a and c).

Subtracting a from c,

[latex]100x - x = 35.\overline{35} - 0.\overline{35}[/latex]

[latex]99x = 35[/latex]

Dividing both sides by 99,

[latex]\displaystyle{x = \frac{35}{99}}[/latex]

Therefore, [latex]\displaystyle{0.353535... = \frac{35}{99}}[/latex].

From Examples 2.5-a to 2.5-d, we have learned that it is possible to convert terminating decimal numbers (e.g., 0.015) and repeating, non-terminating decimal numbers (e.g., 0.65) into fractions. Therefore, such decimal numbers are rational numbers.

However, it is not possible to convert non-repeating, non-terminating decimal numbers (e.g., [latex]\sqrt{2} , \pi, 5.81271...[/latex]) to fractions. Such decimal numbers are referred to as irrational numbers.

These rational numbers and irrational (non-rational) numbers together form the real numbers in the number system.

Converting Fractions to Decimal Numbers

Converting Proper and Improper Fractions to Decimal Numbers

A proper or improper fraction can be converted to its equivalent decimal number by dividing the numerator by the denominator, as shown in the following examples.

Example 2.5-e: Converting Proper and Improper Fractions to Decimal Numbers

Convert the following fractions to their decimal number equivalents:

  1. [latex]\displaystyle{\frac{3}{8}}[/latex]
  2. [latex]\displaystyle{\frac{15}{11}}[/latex]

Solution

  1. [latex]\displaystyle{\frac{3}{8} = 3 \div 8 = 0.375}[/latex]

    Therefore, 0.375 is the decimal equivalent of [latex]\displaystyle{\frac{3}{8}}[/latex].

  2. [latex]\displaystyle{\frac{15}{11} = 15 \div 11 = = 1.363636.... = 1.36}[/latex]

    Therefore, 1.36 is the decimal equivalent of [latex]\displaystyle{\frac{15}{11}}[/latex].

Converting Mixed Numbers to Decimal Numbers

A mixed number can be converted to its decimal form by first converting it to an improper fraction, then dividing the numerator by the denominator, as shown in the following example.

Example 2.5-f: Converting Mixed Numbers to Decimal Numbers

Convert the following mixed numbers to their decimal number equivalents:

  1. [latex]\displaystyle{3\frac{2}{5}}[/latex]
  2. [latex]\displaystyle{11\frac{3}{7}}[/latex]

Solution

  1. [latex]\displaystyle{3\frac{2}{5}}[/latex]

    Coverting to an improper fraction,

    [latex]\displaystyle{= 3\frac{3(5) + 2}{5} = \frac{17}{5}}[/latex]

    Dividing the numerator by the denominator,

    [latex]= 3.4[/latex]

    Therefore, the decimal number equivalent of [latex]\displaystyle{3\frac{2}{5}}[/latex] is [latex]= 3.4[/latex].

  2. [latex]\displaystyle{11\frac{3}{7}}[/latex]

    Coverting to an improper fraction,

    [latex]\displaystyle{= 3\frac{11(7) + 3}{7} = \frac{80}{7}}[/latex]

    Dividing the numerator by the denominator,

    [latex]= 11.428571... = 11.43[/latex]

    Therefore, the decimal number equivalent of [latex]\displaystyle{11\frac{3}{7}}[/latex], rounded to two decimal places, is [latex]11.43[/latex].

Combined Order of Operations

The Order of Operations (BEDMAS), as learned in Chapter 1, Section 1.4, is also used in evaluating expressions with fractions and decimal numbers.

Arithmetic expressions with fractions and decimal numbers that contain multiple operations are performed in the following sequence:

  1. Evaluate the expressions within grouping symbols (i.e., Brackets and radical signs).
  2. Evaluate powers (i.e., Exponents) and roots.
  3. Perform Division and Multiplication, in order from left to right.
  4. Perform Addition and Subtraction, in order from left to right.

Note: For multiplication, division, powers, and roots of mixed numbers, they must first be converted to improper fractions before proceeding with the Order of Operations.

Example 2.5-g: Evaluating Expressions Using Order of Operations (BEDMAS)

Evaluate the following expressions:

  1. [latex]\displaystyle{\left(1\frac{1}{3}\right)^2 + \sqrt{\frac{5}{16} + \frac{20}{16}}}[/latex]
  2. [latex]\displaystyle{\left(\frac{2}{3}\right)^2 + \frac{1}{2}\left(4\frac{1}{2}\right)^2 \div \sqrt{81}}[/latex]
  3. [latex]\displaystyle{\left(\frac{4}{5}\right)^2 + \left(\frac{11}{9} + \sqrt{\frac{49}{81}}\right) \times \frac{3}{25}}[/latex]
  4. [latex]\displaystyle{\sqrt{1\frac{69}{100}} + \sqrt{0.09} + \sqrt{\frac{64}{25}}}[/latex]
  5. [latex]\displaystyle{\left(1 + \frac{0.08}{4}\right)^2 - 1}[/latex]

Solution

  1. [latex]\displaystyle{\left(1\frac{1}{3}\right)^2 + \sqrt{\frac{5}{16} + \frac{20}{16}} = \left(\frac{4}{3}\right)^2 + \sqrt{\frac{25}{16}} = \left(\frac{4}{3}\right)\left(\frac{4}{3}\right) + \frac{\sqrt{25}}{\sqrt{16}}}[/latex][latex]\displaystyle{= \frac{16}{9} + \frac{5}{4} = \frac{64}{36} + \frac{45}{36} = \frac{109}{36} = 3\frac{1}{36}}[/latex]
  2. [latex]\displaystyle{\left(\frac{2}{3}\right)^2 + \frac{1}{2}\left(4\frac{1}{2}\right)^2 \div \sqrt{81} = \left(\frac{2}{3}\right)\left(\frac{2}{3}\right) + \frac{1}{2}\left(\frac{9}{2}\right)\left(\frac{9}{2}\right) \div 9 = \frac{4}{9} + \frac{81}{8} \times \frac{1}{9}}[/latex][latex]\displaystyle{= \frac{4}{9} + \frac{9}{8} = \frac{32}{72} + \frac{81}{72} = \frac{113}{72} = 1\frac{41}{72}}[/latex]
  3. [latex]\displaystyle{\left(\frac{4}{5}\right)^2 + \left(\frac{11}{9} + \sqrt{\frac{49}{81}}\right) \times \frac{3}{25} = \left(\frac{4}{5}\right)^2 + \left(\frac{11}{9} + \frac{\sqrt{49}}{\sqrt{81}}\right) \times \frac{3}{25} = \left(\frac{4}{5}\right)^2 + \left(\frac{11}{9} + \frac{7}{9}\right) \times \frac{3}{25}}[/latex][latex]\displaystyle{= \left(\frac{4}{5}\right)^2 + \frac{18}{9} \times \frac{3}{25} = \left(\frac{4}{5}\right)\left(\frac{4}{5}\right) + 2 \times \frac{3}{25} = \frac{16}{25} + \frac{6}{25} = \frac{22}{25}}[/latex]
  4. [latex]\displaystyle{\sqrt{1\frac{69}{100}} + \sqrt{0.09} + \sqrt{\frac{64}{25}} = \sqrt{\frac{169}{100}} + \sqrt{\frac{9}{100}} + \sqrt{\frac{64}{25}} = \frac{\sqrt{169}}{\sqrt{100}} + \frac{\sqrt{9}}{\sqrt{100}} + \frac{\sqrt{64}}{\sqrt{25}}}[/latex][latex]\displaystyle{= \frac{13}{10} + \frac{3}{10} + \frac{8}{5} = \frac{13}{10} + \frac{3}{10} + \frac{16}{10} = \frac{32}{10} = \frac{16}{5} = 3\frac{1}{5}}[/latex]
  5. [latex]\displaystyle{\left(1 + \frac{0.08}{4}\right)^2 - 1 = \left(1 + 0.02\right)^2 - 1 = \left(1.02\right)^2 - 1}[/latex][latex]\displaystyle{= \left(1.02\right)\left(1.02\right) - 1 = 1.0404 - 1 = 0.0404}[/latex]

Example 2.5-h: Evaluating Expressions by Using the Order of Operations (BEDMAS)

Evaluate: [latex]\displaystyle{\frac{4}{2^3}[(0.5 \times 5^2 + 2.5)^2 \div 3^2] + \sqrt{25}}[/latex]

Solution

[latex]\displaystyle{\frac{4}{2^3}[(0.5 \times 5^2 + 2.5)^2 \div 3^2] + \sqrt{25}}[/latex]

[latex]\displaystyle{= \frac{4}{2^3}[(0.5 \times 25 + 2.5)^2 \div 3^2] + \sqrt{25}}[/latex]

[latex]\displaystyle{= \frac{4}{2^3}[(12.5 + 2.5)^2 \div 3^2] + \sqrt{25}}[/latex]

[latex]\displaystyle{= \frac{4}{2^3}[15^2 \div 3^2] + \sqrt{25}}[/latex]

[latex]\displaystyle{= \frac{4}{2^3}[225 \div 9] + \sqrt{25}}[/latex]

[latex]\displaystyle{= \frac{4}{2^3} \times 25 + \sqrt{25}}[/latex]

[latex]\displaystyle{= \frac{4}{8} \times 25 + 5}[/latex]

[latex]\displaystyle{= 0.5 \times 25 + 5}[/latex]

[latex]\displaystyle{= 12.5 + 5}[/latex]

[latex]\displaystyle{= 17.5}[/latex]

2.5 Exercises

Answers to the odd-numbered problems are available at the end of the textbook.

For problems 1 to 8, convert the decimal numbers to proper fractions and the proper fractions to decimal numbers indicated by the question marks in the tables below.

  1. Question Decimal Number Proper Fraction
    a. 0.2 ?
    b. ? [latex]\displaystyle{\frac{3}{4}}[/latex]
    c. 0.06 ?
  2. Question Decimal Number Proper Fraction
    a. 0.26 ?
    b. ? [latex]\displaystyle{\frac{41}{50}}[/latex]
    c. 0.92 ?
  3. Question Decimal Number Proper Fraction
    a. ? [latex]\displaystyle{\frac{9}{25}}[/latex]
    b. 0.004 ?
    c. ? [latex]\displaystyle{\frac{7}{50}}[/latex]
  4. Question Decimal Number Proper Fraction
    a. ? [latex]\displaystyle{\frac{16}{25}}[/latex]
    b. 0.225 ?
    c. ? [latex]\displaystyle{\frac{19}{20}}[/latex]
  5. Question Decimal Number Proper Fraction
    a. ? [latex]\displaystyle{\frac{1}{2}}[/latex]
    b. 0.4 ?
    c. ? [latex]\displaystyle{\frac{3}{50}}[/latex]
  6. Question Decimal Number Proper Fraction
    a. ? [latex]\displaystyle{\frac{13}{50}}[/latex]
    b. 0.425 ?
    c. ? [latex]\displaystyle{\frac{14}{25}}[/latex]
  7. Question Decimal Number Proper Fraction
    a. 0.005 ?
    b. ? [latex]\displaystyle{\frac{9}{25}}[/latex]
    c. 0.01 ?
  8. Question Decimal Number Proper Fraction
    a. 0.66 ?
    b. ? [latex]\displaystyle{\frac{43}{50}}[/latex]
    c. 0.78 ?

For problems 9 to 12, convert the decimal numbers to improper fractions and the improper fractions to decimal numbers.

  1. Question Decimal Number Improper Fraction
    a. 3.5 ?
    b. ? [latex]\displaystyle{\frac{8}{5}}[/latex]
    c. 5.6 ?
  2. Question Decimal Number Improper Fraction
    a. 7.2 ?
    b. ? [latex]\displaystyle{\frac{37}{5}}[/latex]
    c. 8.4 ?
  3. Question Decimal Number Improper Fraction
    a. ? [latex]\displaystyle{\frac{101}{20}}[/latex]
    b. 6.8 ?
    c. ? [latex]\displaystyle{\frac{11}{4}}[/latex]
  4. Question Decimal Number Improper Fraction
    a. ? [latex]\displaystyle{\frac{107}{50}}[/latex]
    b. 4.8 ?
    c. ? [latex]\displaystyle{\frac{23}{4}}[/latex]

For problems 13 to 16, convert the decimal numbers to mixed numbers and the mixed numbers to decimal numbers.

  1. Question Decimal Number Mixed Number
    a. 2.25 ?
    b. ? [latex]\displaystyle{1\frac{3}{4}}[/latex]
    c. 4.02 ?
  2. Question Decimal Number Mixed Number
    a. 5.04 ?
    b. ? [latex]\displaystyle{12\frac{3}{5}}[/latex]
    c. 14.025 ?
  3. Question Decimal Number Mixed Number
    a. ? [latex]\displaystyle{8\frac{7}{20}}[/latex]
    b. 16.005 ?
    c. ? [latex]\displaystyle{15\frac{1}{2}}[/latex]
  4. Question Decimal Number Mixed Number
    a. ? [latex]\displaystyle{3\frac{5}{8}}[/latex]
    b. 4.75 ?
    c. ? [latex]\displaystyle{5\frac{9}{20}}[/latex]

For problems 17 to 20, convert the repeating decimal numbers to proper fractions and the proper fractions to repeating decimal numbers.

  1. Question Decimal Number Proper Fraction
    a. [latex]0.\overline{6}[/latex] ?
    b. ? [latex]\displaystyle{\frac{23}{90}}[/latex]
    c. [latex]0.\overline{25}[/latex] ?
  2. Question Decimal Number Proper Fraction
    a. [latex]0.\overline{27}[/latex] ?
    b. ? [latex]\displaystyle{\frac{7}{4}}[/latex]
    c. [latex]0.\overline{83}[/latex] ?
  3. Question Decimal Number Proper Fraction
    a. ? [latex]\displaystyle{\frac{5}{11}}[/latex]
    b. [latex]0.\overline{2}[/latex] ?
    c. ? [latex]\displaystyle{\frac{2}{7}}[/latex]
  4. Question Decimal Number Proper Fraction
    a. ? [latex]\displaystyle{\frac{4}{99}}[/latex]
    b. [latex]0.\overline{75}[/latex] ?
    c. ? [latex]\displaystyle{\frac{11}{15}}[/latex]

For problems 21 to 52, evaluate the expressions.

  1. a. [latex]\displaystyle{\left(\frac{3}{5}\right)^2\left(\frac{2}{3}\right)^3}[/latex]
    b. [latex]\displaystyle{\left(\frac{3}{4}\right)^3\left(\frac{1}{6}\right)^2}[/latex]
  2. a. [latex]\displaystyle{\left(\frac{5}{2}\right)^3\left(\frac{1}{3}\right)^2}[/latex]
    b. [latex]\displaystyle{\left(\frac{3}{8}\right)^2\left(\frac{4}{3}\right)^3}[/latex]
  3. a. [latex]\displaystyle{\left(\frac{1}{4}\right)^2 \div \left(\frac{2}{3}\right)^3}[/latex]
    b. [latex]\displaystyle{\left(\frac{5}{3}\right)^2 \div \left(\frac{10}{9}\right)^2}[/latex]
  4. a. [latex]\displaystyle{\left(\frac{1}{2}\right)^2 \div \left(\frac{1}{3}\right)^2}[/latex]
    b. [latex]\displaystyle{\left(\frac{2}{3}\right)^2 \div \left(\frac{4}{9}\right)^2}[/latex]
  5. a. [latex]\displaystyle{\sqrt{\frac{5}{9} + \frac{4}{9}}}[/latex]
    b. [latex]\displaystyle{\sqrt{\frac{15}{36} + \frac{5}{18}}}[/latex]
  6. a. [latex]\displaystyle{\sqrt{\frac{2}{25} + \frac{14}{25}}}[/latex]
    b. [latex]\displaystyle{\sqrt{\frac{1}{16} + \frac{1}{2}}}[/latex]
  7. a. [latex]\displaystyle{\left(2\frac{1}{6} + 1\frac{2}{3}\right) \div 5\frac{3}{4}}[/latex]
    b. [latex]\displaystyle{8\frac{1}{2} \div \left(2\frac{2}{5} + 2\right)}[/latex]
  8. a. [latex]\displaystyle{\left(5\frac{1}{4} + 2\frac{5}{6}\right) \div 1\frac{1}{2}}[/latex]
    b. [latex]\displaystyle{2\frac{2}{3} \div \left(1\frac{7}{15} + \frac{2}{3}\right)}[/latex]
  9. a. [latex]\displaystyle{\left(\frac{3}{5}\right)^2 + \left(1\frac{1}{5}\right)(\sqrt{144})}[/latex]
    b. [latex]\displaystyle{\left(\frac{2}{5}\right)^2 + \left(\frac{3}{2}\right)^3}[/latex]
  10. a. [latex]\displaystyle{\left(\frac{4}{7}\right)^2 + \sqrt{\frac{3}{9} + \frac{1}{9}}}[/latex]
    b. [latex]\displaystyle{\left(\frac{3}{8}\right)^2 + \left(\frac{1}{2}\right)^3}[/latex]
  11. a. [latex]\displaystyle{\sqrt{4\frac{21}{25}} \times \left(\frac{5}{3}\right)^2}[/latex]
    b. [latex]\displaystyle{\left(\frac{1}{4}\right)^2 \times \left(\frac{1}{8}\right)^2}[/latex]
  12. a. [latex]\displaystyle{\sqrt{1\frac{9}{16}} \times \left(\frac{4}{5}\right)^2}[/latex]
    b. [latex]\displaystyle{\left(\frac{1}{3}\right)^2 \div \left(\frac{1}{6}\right)^2}[/latex]
  13. a. [latex](1.3)^2 \times \sqrt{0.04}[/latex]
    b. [latex]\displaystyle{(0.1)^3 \div \sqrt{\frac{1}{100}}}[/latex]
  14. a. [latex](0.01)^2 \times \sqrt{0.09}[/latex]
    b. [latex]\displaystyle{(0.5)^3 \div \sqrt{\frac{1}{100}}}[/latex]
  15. [latex]\displaystyle{\left(\frac{5}{8}\right)^2 + \frac{3}{16} + \frac{5}{12} + 1\frac{2}{3}}[/latex]
  16. [latex]\displaystyle{\left(\frac{6}{9}\right)^2 + 1\frac{5}{9} + \frac{5}{6} + 4\frac{1}{2}}[/latex]
  17. [latex]\displaystyle{\sqrt{\frac{7}{9} - \frac{2}{3}} \div \left(\frac{1}{12} + \frac{1}{9}\right)}[/latex]
  18. [latex]\displaystyle{\left(\frac{5}{12} + \frac{3}{8}\right) \div \sqrt{\frac{4}{18} + \frac{1}{36}}}[/latex]
  19. [latex]\displaystyle{10\frac{1}{2} \div 4\frac{1}{5} + \frac{9}{10} \times 2\frac{2}{5} - \frac{3}{7}}[/latex]
  20. [latex]\displaystyle{7\frac{2}{3} \div 2\frac{1}{3} + \frac{3}{5} \times 4\frac{2}{3} - \frac{2}{5}}[/latex]
  21. [latex]\displaystyle{13\frac{3}{7} \times \frac{5}{94} + 6\frac{4}{5} \div \frac{4}{15} + 7\frac{1}{5}}[/latex]
  22. [latex]\displaystyle{2\frac{1}{6} \div \frac{26}{45} + 3\frac{1}{4} \times 4\frac{1}{2} + \frac{3}{8}}[/latex]
  23. [latex][0.8 - (7.2 - 6.5)] \div [3 \div (3.4 - 0.4)][/latex]
  24. [latex](9.9 \div 1.1) \div (8.1 \div 1.5) + (9.2 - 7.7 + 1.5)[/latex]
  25. [latex](9.2 + 2.8) 0.25 \div (5.6 - 2.3 + 1.7)[/latex]
  26. [latex](9.1 - 7.3) 0.5 \div (5.8 + 8.6 - 5.4)[/latex]
  27. [latex]\displaystyle{\sqrt{49} + \frac{8}{4^2}(0.4 \times 6^2 \div 1.2)^2}[/latex]
  28. [latex]\displaystyle{\sqrt{81} + \frac{12.5}{5^2}(0.3 \times 5^2 \div 1.5)^2}[/latex]
  29. [latex]\displaystyle{\sqrt{2\frac{25}{100}} + \sqrt{0.16} + \sqrt{\frac{49}{16}}}[/latex]
  30. [latex]\displaystyle{\sqrt{1\frac{21}{100}} + \sqrt{0.25} + \sqrt{\frac{25}{36}}}[/latex]
  31. [latex]\displaystyle{\frac{9}{50}\left(\sqrt{\frac{49}{81}} + 1\frac{4}{9}\right) + \left(1\frac{3}{5}\right)^2}[/latex]
  32. [latex]\displaystyle{\frac{16}{25}\left(\sqrt{\frac{25}{16}} + 2\frac{1}{4}\right) + \left(1\frac{2}{5}\right)^2}[/latex]

Unless otherwise indicated, this chapter is an adaptation of the eTextbook Foundations of Mathematics (3rd ed.) by Thambyrajah Kugathasan, published by Vretta-Lyryx Inc., with permission. Adaptations include supplementing existing material and reordering chapters.

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