Class Notes: Converting Decimals to Fractions

Objective

Learn how to convert decimals (rounded to the nearest 10th, 100th, 1000th, etc.) into fractions and how to reduce them to their simplest form.

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Key Steps to Convert Decimals to Fractions

1. Identify the Decimal Place

   • 1 digit after decimal: 10th place
   • 2 digits after decimal: 100th place
   • 3 digits after decimal: 1000th place

2. Set up the Fraction

   • Put the decimal over 1.
   • Multiply the numerator and the denominator by 10 for each decimal place to move the decimal point to the right.

3. Simplify the Fraction

   • Factor the numerator and denominator if possible.
   • Cancel any common factors.
   • Continue simplifying until the fraction can no longer be reduced.

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Examples

Example 1: Decimal in the 10th Place

• Decimal: 0.3

  • Multiply by 10:
    • (0.3 \times 10 = 3)
    • (1 \times 10 = 10)
  • Fraction: ( \frac{3}{10} )

• Decimal: 0.4

  • Multiply by 10:
    • (0.4 \times 10 = 4)
    • (1 \times 10 = 10)
  • Fraction before simplifying: ( \frac{4}{10} )
  • Simplified fraction: ( \frac{2}{5} )

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Example 2: Decimal in the 100th Place

• Decimal: 0.32

  • Multiply by 100:
    • (0.32 \times 100 = 32)
    • (1 \times 100 = 100)
  • Fraction before simplifying: ( \frac{32}{100} )
  • Reduce by dividing numerator and denominator by 4 or by successive division by 2:
    • (32 / 2 = 16,\ 100 / 2 = 50)
    • (16 / 2 = 8,\ 50 / 2 = 25)
    • Final fraction: ( \frac{8}{25} )

• Decimal: 2.25

  • Multiply by 100:
    • (2.25 \times 100 = 225)
    • (1 \times 100 = 100)
  • Fraction before simplifying: ( \frac{225}{100} )
  • Simplify by dividing by 5 twice:
    • (225 / 5 = 45,\ 100 / 5 = 20)
    • (45 / 5 = 9,\ 20 / 5 = 4)
    • Final fraction: ( \frac{9}{4} ) (improper fraction; can be written as a mixed number if desired)

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Example 3: Decimal in the 1000th Place

• Decimal: 0.128
  • Multiply by 1000:
    • (0.128 \times 1000 = 128)
    • (1 \times 1000 = 1000)
  • Fraction before simplifying: ( \frac{128}{1000} )
  • Simplify by successive division by 2:
    • (128 / 2 = 64,\ 1000 / 2 = 500)
    • (64 / 2 = 32,\ 500 / 2 = 250)
    • (32 / 2 = 16,\ 250 / 2 = 125)
    • Final fraction: ( \frac{16}{125} )

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Summary Table

| Decimal | Place Value | Multiply by | Numerator | Denominator | Simplified Fraction | |---------|------------|-------------|-----------|-------------|---------------------| | 0.3 | 10th | 10 | 3 | 10 | 3/10 | | 0.4 | 10th | 10 | 4 | 10 | 2/5 | | 0.32 | 100th | 100 | 32 | 100 | 8/25 | | 2.25 | 100th | 100 | 225 | 100 | 9/4 | | 0.128 | 1000th | 1000 | 128 | 1000 | 16/125 |

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Tips for Simplifying Fractions

• Check for common prime factors in numerator and denominator.
• If both are even, keep dividing by 2.
• If both end in 0 or 5, divide by 5.
• Continue until no further reduction is possible.

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General Rule

• For decimals in the 10th place: Multiply top and bottom by 10.
• For decimals in the 100th place: Multiply top and bottom by 100.
• For decimals in the 1000th place: Multiply top and bottom by 1,000.
• Simplify the resulting fraction as much as possible.