To solve this problem, we need to determine how many pieces of length [tex]\(3 \frac{1}{3}\)[/tex] inches can be cut from a licorice rope that is [tex]\(30 \frac{2}{3}\)[/tex] inches long.
1. Convert Mixed Numbers to Improper Fractions:
Let's first convert the mixed numbers into improper fractions for easier arithmetic operations.
- Total length of the licorice rope:
[tex]\[
30 \frac{2}{3} = 30 + \frac{2}{3} = \frac{30 \times 3}{3} + \frac{2}{3} = \frac{90}{3} + \frac{2}{3} = \frac{92}{3}
\][/tex]
- Length of each piece:
[tex]\[
3 \frac{1}{3} = 3 + \frac{1}{3} = \frac{3 \times 3}{3} + \frac{1}{3} = \frac{9}{3} + \frac{1}{3} = \frac{10}{3}
\][/tex]
2. Divide the Total Length by the Length of Each Piece:
Now, we need to divide the total length of the rope by the length of each piece to determine how many pieces we can cut.
[tex]\[
\text{Number of pieces} = \frac{\text{Total length}}{\text{Length of each piece}} = \frac{\frac{92}{3}}{\frac{10}{3}}
\][/tex]
To divide fractions, we multiply the first fraction by the reciprocal of the second fraction:
[tex]\[
\text{Number of pieces} = \frac{\frac{92}{3}}{\frac{10}{3}} = \frac{92}{3} \times \frac{3}{10} = \frac{92 \times 3}{3 \times 10} = \frac{276}{30}
\][/tex]
Simplify the fraction by dividing the numerator and the denominator by their greatest common divisor (GCD), which is 6:
[tex]\[
\frac{276 \div 6}{30 \div 6} = \frac{46}{5} = 9 \frac{1}{5}
\][/tex]
3. Conclusion:
Therefore, when Marissa cuts the [tex]\(30 \frac{2}{3}\)[/tex] inch licorice rope into pieces that are [tex]\(3 \frac{1}{3}\)[/tex] inches each, she will get exactly [tex]\(9 \frac{1}{5}\)[/tex] pieces.
Therefore, the answer is:
[tex]\[ \boxed{9 \frac{1}{5}} \][/tex]
