Multiplying Mixed Numbers

Key Concept

To multiply mixed numbers, convert them to improper fractions and multiply. This avoids the complexity of the distributive property.

Steps

1. Convert Mixed Numbers to Improper Fractions

   • For a mixed number ( a \frac{b}{c} ):
     • Multiply the whole number ( a ) by a fraction with the same denominator as the fractional part: ( a \times \frac{c}{c} ).
     • Add this product to the fractional part (\frac{b}{c}).
     • Example: Convert ( 4 \frac{1}{2} ) to improper fraction: [ 4 \times \frac{2}{2} + \frac{1}{2} = \frac{8}{2} + \frac{1}{2} = \frac{9}{2} ]

2. Multiply the Improper Fractions

   • Multiply the numerators and the denominators.
   • Example: [ \frac{9}{2} \times \frac{5}{3} = \frac{9 \times 5}{2 \times 3} = \frac{45}{6} ]

3. Convert the Product Back to a Mixed Number

   • Divide the numerator by the denominator to get the whole number.
   • The remainder becomes the numerator of the fraction part.
   • Example: [ 45 \div 6 = 7 , \text{with remainder} , 3 \quad \Rightarrow , 7 \frac{3}{6} = 7 \frac{1}{2} ]

Example Problems

1. Example 1: ( 4 \frac{1}{2} \times 1 \frac{2}{3} )

   • Convert ( 4 \frac{1}{2} ) to ( \frac{9}{2} )
   • Convert ( 1 \frac{2}{3} ) to ( \frac{5}{3} )
   • Multiply: [ \frac{9}{2} \times \frac{5}{3} = \frac{45}{6} = 7 \frac{1}{2} ]

2. Example 2: ( 2 \frac{1}{3} \times 3 \frac{3}{4} )

   • Convert ( 2 \frac{1}{3} ) to ( \frac{7}{3} )
   • Convert ( 3 \frac{3}{4} ) to ( \frac{15}{4} )
   • Multiply: [ \frac{7}{3} \times \frac{15}{4} = \frac{105}{12} = 8 \frac{3}{4} ]

Conclusion

By converting mixed numbers to improper fractions, multiplying becomes straightforward. To practice, work on problems independently to reinforce these steps.

Further Practice

Keep practicing with more problems, and access additional printable worksheets for extra exercises.