How to Convert Between Fractions, Decimals and Mixed Numbers

How to Convert between Fractions, Decimals and Mixed Numbers

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Fractions, decimals, and mixed numbers are different ways to write the same value. A fraction compares a numerator to a denominator, a decimal uses place value based on powers of \(10\), and a mixed number combines a whole number with a proper fraction.

To change a fraction to a decimal, divide the numerator by the denominator or rewrite the denominator as \(10\), \(100\), \(1000\), and so on when possible. To change a decimal to a fraction, use its place value, then simplify. To change between mixed numbers and improper fractions, use \(a\frac{b}{c}=\frac{ac+b}{c}\).

Examples

Example 1: \(0.48=\frac{48}{100}=\frac{12}{25}\).

Example 2: \(3\frac{2}{5}=3+\frac{2}{5}=3+0.4=3.4\).

Example 3: \(\frac{19}{6}=3\frac{1}{6}\), because \(19=6\cdot 3+1\).

Practice Exercises

1) Write \(\frac{3}{10}\) as a decimal.

2) Write \(0.7\) as a fraction in simplest form.

3) Write \(\frac{5}{4}\) as a mixed number.

4) Write \(2\frac{1}{2}\) as an improper fraction.

5) Write \(0.35\) as a fraction in simplest form.

6) Write \(\frac{7}{8}\) as a decimal.

7) Write \(3.125\) as a mixed number in simplest form.

8) Write \(4\frac{3}{5}\) as a decimal.

9) Write \(0.006\) as a fraction in simplest form.

10) Write \(\frac{11}{20}\) as a decimal.

11) Write \(5.04\) as a mixed number in simplest form.

12) Write \(\frac{17}{6}\) as a mixed number.

13) Write \(6\frac{7}{8}\) as a decimal.

14) Write \(0.375\) as a fraction in simplest form.

15) Write \(\frac{13}{25}\) as a decimal.

16) Write \(12.125\) as an improper fraction.

17) Write \(0.\overline{3}\) as a fraction.

18) Write \(2.\overline{16}\) as a mixed number.

19) Order \(\frac{3}{4},\ 0.72,\ 0.705,\ 1\frac{1}{5}\) from least to greatest.

20) A recipe uses \(1.75\) cups of flour. Write this amount as a mixed number and as an improper fraction.

Answers

1) Tenths become one digit after the decimal point. \(\frac{3}{10}=0.3\).

2) \(0.7\) means seven tenths, so \(0.7=\frac{7}{10}\). Since \(7\) and \(10\) have no common factor greater than \(1\), it is simplest.

3) Divide \(5\) by \(4\): \(5=4\cdot 1+1\). Therefore \(\frac{5}{4}=1\frac{1}{4}\).

4) Multiply the whole number by the denominator and add the numerator: \(2\cdot 2+1=5\). Keep the denominator \(2\). So \(2\frac{1}{2}=\frac{5}{2}\).

5) \(0.35=\frac{35}{100}\). Divide numerator and denominator by \(5\): \(\frac{35}{100}=\frac{7}{20}\). Answer: \(\frac{7}{20}\).

6) Divide \(7\) by \(8\): \(7.000\div 8=0.875\). Therefore \(\frac{7}{8}=0.875\).

7) \(3.125=3+\frac{125}{1000}\). Simplify \(\frac{125}{1000}\) by dividing by \(125\): \(\frac{1}{8}\). Answer: \(3\frac{1}{8}\).

8) Convert the fraction part: \(\frac{3}{5}=0.6\). Then \(4\frac{3}{5}=4.6\).

9) \(0.006=\frac{6}{1000}\). Divide by \(2\): \(\frac{6}{1000}=\frac{3}{500}\). Answer: \(\frac{3}{500}\).

10) Make the denominator \(100\): \(\frac{11}{20}=\frac{55}{100}\). Therefore the decimal is \(0.55\).

11) \(5.04=5+\frac{4}{100}\). Simplify \(\frac{4}{100}=\frac{1}{25}\). Answer: \(5\frac{1}{25}\).

12) Divide \(17\) by \(6\): \(17=6\cdot 2+5\). Therefore \(\frac{17}{6}=2\frac{5}{6}\).

13) Convert \(\frac{7}{8}\) by division: \(7\div 8=0.875\). Add the whole number: \(6\frac{7}{8}=6.875\).

14) \(0.375=\frac{375}{1000}\). Divide by \(125\): \(\frac{375}{1000}=\frac{3}{8}\). Answer: \(\frac{3}{8}\).

15) Make the denominator \(100\): \(\frac{13}{25}=\frac{52}{100}\). Therefore \(\frac{13}{25}=0.52\).

16) \(12.125=12+\frac{125}{1000}=12+\frac{1}{8}\). Convert to an improper fraction: \(12\frac{1}{8}=\frac{12\cdot 8+1}{8}=\frac{97}{8}\).

17) Let \(x=0.\overline{3}\). Then \(10x=3.\overline{3}\). Subtract: \(10x-x=3\), so \(9x=3\). Thus \(x=\frac{3}{9}=\frac{1}{3}\).

18) Let the repeating part be \(0.\overline{16}\). If \(x=0.\overline{16}\), then \(100x=16.\overline{16}\). Subtract \(x\): \(99x=16\), so \(x=\frac{16}{99}\). Therefore \(2.\overline{16}=2\frac{16}{99}\).

19) Convert to decimals: \(\frac{3}{4}=0.75\) and \(1\frac{1}{5}=1.2\). Compare \(0.705,\ 0.72,\ 0.75,\ 1.2\). Least to greatest: \(0.705,\ 0.72,\ \frac{3}{4},\ 1\frac{1}{5}\).

20) \(1.75=1+\frac{75}{100}\). Simplify \(\frac{75}{100}=\frac{3}{4}\), so the mixed number is \(1\frac{3}{4}\). Convert to an improper fraction: \(1\frac{3}{4}=\frac{1\cdot 4+3}{4}=\frac{7}{4}\).

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