1. Introduction
Fractions and decimals are two common ways of representing non-integer numbers. Understanding how to convert between fractions and decimals, as well as performing operations with them, is essential for many areas of mathematics. In these notes, we cover the basic definitions, conversion methods, and operations, along with 50 examples (with solutions) to illustrate the concepts.
2. Basic Definitions
Fractions
A fraction represents a part of a whole and is written in the form:
\( \frac{a}{b} \quad (b \neq 0) \)
where \( a \) is the numerator and \( b \) is the denominator.
Decimals
A decimal is another way of expressing fractions, particularly those with denominators that are powers of 10. For example:
\( 0.75 \)
represents the fraction \( \frac{75}{100} \), which can be simplified to \( \frac{3}{4} \).
3. Converting Between Fractions and Decimals
Fraction to Decimal
To convert a fraction to a decimal, divide the numerator by the denominator.
\(\frac{a}{b} = a \div b\)
Decimal to Fraction
To convert a decimal to a fraction, write the decimal over the appropriate power of 10 and then simplify.
0.125 = \(\frac{125}{1000}\) = \(\frac{1}{8}\)
4. Operations with Fractions and Decimals
Operations with Fractions
Addition/Subtraction: Find a common denominator, convert fractions, and then add/subtract.
Multiplication: Multiply the numerators and denominators:
\(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\)
Division: Multiply by the reciprocal:
\(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\)
Operations with Decimals
Operations with decimals are similar to whole numbers, but it is important to align the decimal points for addition and subtraction. Multiplication and division follow similar rules to whole numbers, with adjustments for the decimal places.
5. 50 Examples with Solutions
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Example 1: Converting Fraction to Decimal
Convert \( \frac{1}{2} \) to a decimal.
Solution: \( \frac{1}{2} = 1 \div 2 = 0.5 \) -
Example 2: Converting Fraction to Decimal
Convert \( \frac{3}{4} \) to a decimal.
Solution: \( \frac{3}{4} = 3 \div 4 = 0.75 \) -
Example 3: Converting Fraction to Decimal
Convert \( \frac{1}{4} \) to a decimal.
Solution: \( \frac{1}{4} = 1 \div 4 = 0.25 \) -
Example 4: Converting Fraction to Decimal
Convert \( \frac{2}{5} \) to a decimal.
Solution: \( \frac{2}{5} = 2 \div 5 = 0.4 \) -
Example 5: Converting Fraction to Decimal
Convert \( \frac{3}{5} \) to a decimal.
Solution: \( \frac{3}{5} = 3 \div 5 = 0.6 \) -
Example 6: Converting Fraction to Decimal
Convert \( \frac{1}{8} \) to a decimal.
Solution: \( \frac{1}{8} = 1 \div 8 = 0.125 \) -
Example 7: Converting Fraction to Decimal
Convert \( \frac{3}{8} \) to a decimal.
Solution: \( \frac{3}{8} = 3 \div 8 = 0.375 \) -
Example 8: Converting Fraction to Decimal
Convert \( \frac{5}{8} \) to a decimal.
Solution: \( \frac{5}{8} = 5 \div 8 = 0.625 \) -
Example 9: Converting Fraction to Decimal
Convert \( \frac{7}{8} \) to a decimal.
Solution: \( \frac{7}{8} = 7 \div 8 = 0.875 \) -
Example 10: Converting Fraction to Decimal
Convert \( \frac{1}{10} \) to a decimal.
Solution: \( \frac{1}{10} = 1 \div 10 = 0.1 \) -
Example 11: Converting Fraction to Decimal
Convert \( \frac{3}{10} \) to a decimal.
Solution: \( \frac{3}{10} = 3 \div 10 = 0.3 \) -
Example 12: Converting Fraction to Decimal
Convert \( \frac{7}{10} \) to a decimal.
Solution: \( \frac{7}{10} = 7 \div 10 = 0.7 \) -
Example 13: Adding Fractions
Compute \( \frac{1}{2} + \frac{1}{4} \).
Solution: Find a common denominator: \( \frac{1}{2} = \frac{2}{4} \), so \( \frac{2}{4} + \frac{1}{4} = \frac{3}{4} \). In decimal, \( \frac{3}{4} = 0.75 \). -
Example 14: Subtracting Fractions
Compute \( \frac{3}{4} - \frac{1}{2} \).
Solution: \( \frac{3}{4} - \frac{2}{4} = \frac{1}{4} = 0.25 \). -
Example 15: Multiplying Fractions
Compute \( \frac{1}{2} \times \frac{2}{3} \).
Solution: \( \frac{1 \times 2}{2 \times 3} = \frac{2}{6} = \frac{1}{3} \approx 0.333 \). -
Example 16: Dividing Fractions
Compute \( \frac{3}{4} \div \frac{1}{2} \).
Solution: Multiply by the reciprocal: \( \frac{3}{4} \times \frac{2}{1} = \frac{6}{4} = \frac{3}{2} = 1.5 \). -
Example 17: Simplifying a Fraction
Simplify \( \frac{8}{12} \).
Solution: Divide numerator and denominator by 4: \( \frac{8\div4}{12\div4} = \frac{2}{3} \approx 0.667 \). -
Example 18: Converting a Repeating Decimal to Fraction
Express \( 0.\overline{3} \) as a fraction.
Solution: \( 0.\overline{3} = \frac{1}{3} \). -
Example 19: Converting a Repeating Decimal to Fraction
Express \( 0.\overline{6} \) as a fraction.
Solution: \( 0.\overline{6} = \frac{2}{3} \). -
Example 20: Adding Fractions with Different Denominators
Compute \( \frac{1}{3} + \frac{1}{4} \).
Solution: Common denominator is 12: \( \frac{4}{12} + \frac{3}{12} = \frac{7}{12} \approx 0.5833 \). -
Example 21: Subtracting Fractions with Different Denominators
Compute \( \frac{5}{6} - \frac{1}{4} \).
Solution: Common denominator is 12: \( \frac{10}{12} - \frac{3}{12} = \frac{7}{12} \approx 0.5833 \). -
Example 22: Multiplying Mixed Numbers
Compute \( 1\frac{1}{2} \times 2\frac{1}{3} \).
Solution: Convert to improper fractions: \( \frac{3}{2} \times \frac{7}{3} = \frac{21}{6} = 3.5 \). -
Example 23: Dividing Mixed Numbers
Compute \( 2\frac{1}{2} \div 1\frac{1}{4} \).
Solution: Convert: \( \frac{5}{2} \div \frac{5}{4} = \frac{5}{2} \times \frac{4}{5} = 2 \). -
Example 24: Converting a Mixed Number to Decimal
Convert \( 3\frac{1}{4} \) to a decimal.
Solution: \( 3\frac{1}{4} = \frac{13}{4} = 3.25 \). -
Example 25: Decimal Addition
Compute \( 0.5 + 0.75 \).
Solution: \( 0.5 + 0.75 = 1.25 \). -
Example 26: Decimal Subtraction
Compute \( 1.25 - 0.5 \).
Solution: \( 1.25 - 0.5 = 0.75 \). -
Example 27: Decimal Multiplication
Compute \( 0.5 \times 0.4 \).
Solution: \( 0.5 \times 0.4 = 0.20 \). -
Example 28: Decimal Division
Compute \( 0.75 \div 0.25 \).
Solution: \( 0.75 \div 0.25 = 3 \). -
Example 29: Converting a Repeating Decimal
Express \( 0.\overline{81} \) as a fraction.
Solution: \( 0.\overline{81} = \frac{81}{99} = \frac{9}{11} \approx 0.8181 \). -
Example 30: Converting a Decimal to a Fraction
Convert \( 0.125 \) to a fraction.
Solution: \( 0.125 = \frac{125}{1000} = \frac{1}{8} \). -
Example 31: Adding Fractions
Compute \( \frac{2}{5} + \frac{1}{10} \).
Solution: \( \frac{2}{5} = \frac{4}{10} \) and \( \frac{4}{10} + \frac{1}{10} = \frac{5}{10} = \frac{1}{2} = 0.5 \). -
Example 32: Subtracting Fractions
Compute \( \frac{7}{10} - \frac{2}{5} \).
Solution: \( \frac{2}{5} = \frac{4}{10} \) so \( \frac{7}{10} - \frac{4}{10} = \frac{3}{10} = 0.3 \). -
Example 33: Multiplying Fractions
Compute \( \frac{3}{5} \times \frac{2}{3} \).
Solution: \( \frac{3 \times 2}{5 \times 3} = \frac{6}{15} = \frac{2}{5} = 0.4 \). -
Example 34: Dividing Fractions
Compute \( \frac{3}{4} \div \frac{3}{8} \).
Solution: \( \frac{3}{4} \times \frac{8}{3} = \frac{24}{12} = 2 \). -
Example 35: Simplifying a Fraction
Simplify \( \frac{15}{35} \).
Solution: Divide numerator and denominator by 5: \( \frac{15\div5}{35\div5} = \frac{3}{7} \approx 0.4286 \). -
Example 36: Converting a Fraction to Decimal
Convert \( \frac{2}{7} \) to a decimal.
Solution: \( \frac{2}{7} \approx 0.2857 \). -
Example 37: Converting a Fraction to Decimal
Convert \( \frac{5}{7} \) to a decimal.
Solution: \( \frac{5}{7} \approx 0.7143 \). -
Example 38: Adding Decimals
Compute \( 0.333 + 0.667 \).
Solution: \( 0.333 + 0.667 = 1.000 \). -
Example 39: Subtracting Decimals
Compute \( 0.875 - 0.125 \).
Solution: \( 0.875 - 0.125 = 0.750 \). -
Example 40: Multiplying Decimals
Compute \( 0.25 \times 0.4 \).
Solution: \( 0.25 \times 0.4 = 0.1 \). -
Example 41: Dividing Decimals
Compute \( 0.6 \div 0.2 \).
Solution: \( 0.6 \div 0.2 = 3 \). -
Example 42: Converting a Mixed Number to Decimal
Convert \( 2\frac{3}{5} \) to a decimal.
Solution: \( 2\frac{3}{5} = \frac{13}{5} = 2.6 \). -
Example 43: Converting a Decimal to a Fraction
Express \( 0.2 \) as a fraction.
Solution: \( 0.2 = \frac{2}{10} = \frac{1}{5} \). -
Example 44: Converting a Decimal to a Fraction
Express \( 0.75 \) as a fraction.
Solution: \( 0.75 = \frac{75}{100} = \frac{3}{4} \). -
Example 45: Converting a Fraction to Decimal
Convert \( \frac{9}{10} \) to a decimal.
Solution: \( \frac{9}{10} = 9 \div 10 = 0.9 \). -
Example 46: Multiplying Mixed Numbers
Compute \( 1\frac{1}{3} \times 2\frac{2}{3} \).
Solution: Convert to improper fractions: \( \frac{4}{3} \times \frac{8}{3} = \frac{32}{9} \approx 3.56 \). -
Example 47: Dividing Mixed Numbers
Compute \( 3\frac{1}{2} \div 1\frac{2}{3} \).
Solution: Convert to improper fractions: \( \frac{7}{2} \div \frac{5}{3} = \frac{7}{2} \times \frac{3}{5} = \frac{21}{10} = 2.1 \). -
Example 48: Adding Mixed Numbers
Compute \( 1\frac{1}{4} + 2\frac{2}{4} \).
Solution: Convert: \( 1\frac{1}{4} = \frac{5}{4} \) and \( 2\frac{2}{4} = \frac{10}{4} \). Then, \( \frac{5}{4} + \frac{10}{4} = \frac{15}{4} = 3.75 \). -
Example 49: Subtracting Mixed Numbers
Compute \( 4\frac{3}{4} - 2\frac{1}{2} \).
Solution: Convert: \( 4\frac{3}{4} = \frac{19}{4} \) and \( 2\frac{1}{2} = \frac{5}{2} = \frac{10}{4} \). Then, \( \frac{19}{4} - \frac{10}{4} = \frac{9}{4} = 2.25 \). -
Example 50: Converting a Decimal to a Simplified Fraction
Convert \( 0.625 \) to a fraction.
Solution: \( 0.625 = \frac{625}{1000} = \frac{5}{8} \).
6. Summary
In these notes, we reviewed the definitions of fractions and decimals, learned how to convert between the two, and performed various operations such as addition, subtraction, multiplication, and division. The 50 examples provided illustrate a wide range of conversions and operations, ensuring a thorough understanding of fractions and decimals.
