To solve the division of mixed fractions step by step, let's break it down clearly.
### Problem Statement
We need to evaluate:
[tex]\[ 6 \frac{3}{4} \div 2 \frac{1}{3} \][/tex]
Given options:
1. [tex]\( 2 \frac{25}{28} \)[/tex]
2. [tex]\( 3 \frac{1}{3} \)[/tex]
3. [tex]\( 4 \frac{1}{12} \)[/tex]
4. [tex]\( 8 \frac{11}{12} \)[/tex]
5. None of these
### Step-by-Step Solution
1. Convert Mixed Numbers to Improper Fractions:
- [tex]\( 6 \frac{3}{4} \)[/tex]:
[tex]\[ 6 \frac{3}{4} = 6 + \frac{3}{4} = \frac{6 \times 4}{4} + \frac{3}{4} = \frac{24}{4} + \frac{3}{4} = \frac{27}{4} \][/tex]
- [tex]\( 2 \frac{1}{3} \)[/tex]:
[tex]\[ 2 \frac{1}{3} = 2 + \frac{1}{3} = \frac{2 \times 3}{3} + \frac{1}{3} = \frac{6}{3} + \frac{1}{3} = \frac{7}{3} \][/tex]
2. Perform the Division of Improper Fractions:
- To divide by a fraction, multiply by its reciprocal:
[tex]\[ \frac{27}{4} \div \frac{7}{3} = \frac{27}{4} \times \frac{3}{7} = \frac{27 \times 3}{4 \times 7} = \frac{81}{28} \][/tex]
3. Convert the Result to a Mixed Number:
- Divide 81 by 28 to find the whole number and the remainder:
[tex]\[ 81 \div 28 = 2 \text{ remainder } 25 \][/tex]
- Thus, the quotient is 2 and the remainder is 25.
- So, the fraction part is [tex]\( \frac{25}{28} \)[/tex].
- Thus, the mixed number is:
[tex]\[ 2 \frac{25}{28} \][/tex]
4. Compare with the Given Options:
Option 1: [tex]\( 2 \frac{25}{28} \)[/tex] matches the result we calculated.
Option 2: [tex]\( 3 \frac{1}{3} \)[/tex] does not match.
Option 3: [tex]\( 4 \frac{1}{12} \)[/tex] does not match.
Option 4: [tex]\( 8 \frac{11}{12} \)[/tex] does not match.
Option 5: None of these – not applicable here as Option 1 is correct.
### Final Answer
[tex]\[ 6 \frac{3}{4} \div 2 \frac{1}{3} = 2 \frac{25}{28} \][/tex]
Therefore, the correct answer is:
[tex]\[ 2 \frac{25}{28} \][/tex]
