Class Notes: Converting Improper Fractions to Mixed Numbers (Without Long Division)

Objective

• Learn how to convert improper fractions to mixed numbers by breaking down the numerator, avoiding long division.

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Step-by-Step Method

1. Identify the Highest Multiple of the Denominator

• Find the highest multiple of the denominator that is less than or equal to the numerator.
• Break the numerator into two parts:
  • The largest possible multiple of the denominator
  • The remainder

2. Example Walkthroughs

Example 1: ( \frac{11}{4} )

• Find highest multiple of 4 under 11: 8 ((4 \times 2 = 8))
• Break: ( 11 = 8 + 3 )
• Rewrite: ( \frac{11}{4} = \frac{8}{4} + \frac{3}{4} )
• Calculate: ( \frac{8}{4} = 2 )
• Mixed Number: ( 2 + \frac{3}{4} = 2\frac{3}{4} )

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Example 2: ( \frac{17}{5} )

• Highest multiple of 5 under 17: 15 ((5 \times 3 = 15))
• Break: ( 17 = 15 + 2 )
• Rewrite: ( \frac{17}{5} = \frac{15}{5} + \frac{2}{5} )
• Calculate: ( \frac{15}{5} = 3 )
• Mixed Number: ( 3 + \frac{2}{5} = 3\frac{2}{5} )

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Example 3: ( \frac{53}{6} )

• Highest multiple of 6 under 53: 48 ((6 \times 8 = 48))
• Break: ( 53 = 48 + 5 )
• Rewrite: ( \frac{53}{6} = \frac{48}{6} + \frac{5}{6} )
• Calculate: ( \frac{48}{6} = 8 )
• Mixed Number: ( 8 + \frac{5}{6} = 8\frac{5}{6} )

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Example 4: ( \frac{45}{7} )

• Highest multiple of 7 under 45: 42 ((7 \times 6 = 42))
• Break: ( 45 = 42 + 3 )
• Rewrite: ( \frac{45}{7} = \frac{42}{7} + \frac{3}{7} )
• Calculate: ( \frac{42}{7} = 6 )
• Mixed Number: ( 6 + \frac{3}{7} = 6\frac{3}{7} )

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Checking Your Work

• To verify, convert mixed number back to an improper fraction:

  • Multiply whole number by denominator, add numerator of fraction.
  • Keep denominator the same.

  Example: ( 6\frac{3}{7} \rightarrow (6 \times 7) + 3 = 45 \rightarrow \frac{45}{7} )

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Tips

• List out multiples: If unsure, write multiples of the denominator until just below the numerator.
• Subtraction for remainder: Subtract highest multiple from numerator to get the remainder for the fractional part.

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Summary

• No need for long division.
• Break up the numerator using multiples of the denominator.
• Express as whole number plus proper fraction.