How to Divide Two Mixed Numbers

This document serves as a class note summarizing the steps and examples for dividing two mixed numbers.

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Steps for Dividing Mixed Numbers

1. Convert Each Mixed Number to an Improper Fraction

   • For a mixed number ( a ) and ( \frac{b}{c} ), improper fraction = ( \frac{a \times c + b}{c} )

2. Use Keep, Change, Flip

   • Keep: the first fraction stays the same
   • Change: change division to multiplication
   • Flip: flip (take the reciprocal of) the second fraction

3. Multiply the Improper Fractions

4. Convert the Result Back to a Mixed Number

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Example 1

Problem: Divide ( 8 \frac{2}{3} ) by ( 3 \frac{1}{4} )

Step 1: Convert to Improper Fractions

• ( 8 \frac{2}{3} = \frac{8 \times 3 + 2}{3} = \frac{26}{3} )
• ( 3 \frac{1}{4} = \frac{3 \times 4 + 1}{4} = \frac{13}{4} )

Step 2: Apply Keep, Change, Flip

• Keep: ( \frac{26}{3} )
• Change: division to multiplication
• Flip: ( \frac{13}{4} ) to ( \frac{4}{13} )

Calculation: [ \frac{26}{3} \div \frac{13}{4} \rightarrow \frac{26}{3} \times \frac{4}{13} ]

Step 3: Simplify and Multiply

• ( 26 = 13 \times 2 )
• Cancel ( 13 ):

[ \frac{2}{3} \times 4 = \frac{8}{3} ]

Step 4: Convert ( \frac{8}{3} ) to a Mixed Number

• ( 8 \div 3 = 2 ) remainder ( 2 )
• ( \frac{8}{3} = 2 \frac{2}{3} )

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Example 2

Problem: Divide ( 7 \frac{3}{5} ) by ( 3 \frac{1}{2} )

Step 1: Convert to Improper Fractions

• ( 7 \frac{3}{5} = \frac{7 \times 5 + 3}{5} = \frac{38}{5} )
• ( 3 \frac{1}{2} = \frac{3 \times 2 + 1}{2} = \frac{7}{2} )

Step 2: Apply Keep, Change, Flip

• Keep: ( \frac{38}{5} )
• Change: division to multiplication
• Flip: ( \frac{7}{2} ) to ( \frac{2}{7} )

Calculation: [ \frac{38}{5} \div \frac{7}{2} \rightarrow \frac{38}{5} \times \frac{2}{7} ]

Step 3: Multiply

[ 38 \times 2 = 76 \ 5 \times 7 = 35 ] Thus, ( \frac{76}{35} )

Step 4: Convert ( \frac{76}{35} ) to a Mixed Number

• ( 76 \div 35 = 2 ) remainder 6
• ( \frac{76}{35} = 2 \frac{6}{35} )

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Summary

• Step 1: Convert mixed numbers to improper fractions.
• Step 2: Use "keep, change, flip" to rewrite as multiplication.
• Step 3: Multiply.
• Step 4: Convert the answer back to a mixed number.

This process enables you to divide any two mixed numbers.