To multiply and simplify [tex]\(2 \frac{1}{4} \cdot 3 \frac{2}{3}\)[/tex], we follow several steps:
1. Convert the mixed numbers to improper fractions:
- [tex]\(2 \frac{1}{4}\)[/tex]:
[tex]\[2 \frac{1}{4} = 2 + \frac{1}{4} = \frac{8}{4} + \frac{1}{4} = \frac{9}{4}\][/tex]
- [tex]\(3 \frac{2}{3}\)[/tex]:
[tex]\[3 \frac{2}{3} = 3 + \frac{2}{3} = \frac{9}{3} + \frac{2}{3} = \frac{11}{3}\][/tex]
2. Multiply the two improper fractions:
[tex]\[
\frac{9}{4} \cdot \frac{11}{3} = \frac{9 \times 11}{4 \times 3} = \frac{99}{12}
\][/tex]
3. Simplify the resulting fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD):
The GCD of 99 and 12 is 3. Therefore:
[tex]\[
\frac{99}{12} = \frac{99 \div 3}{12 \div 3} = \frac{33}{4}
\][/tex]
4. Convert the simplified improper fraction back to a mixed number:
- Divide the numerator by the denominator to get the whole number part:
[tex]\[33 \div 4 = 8 \text{ R } 1\][/tex]
- The remainder is the numerator of the fractional part.
So,
[tex]\[
\frac{33}{4} = 8 \frac{1}{4}
\][/tex]
Therefore, the final answer is [tex]\(8 \frac{1}{4}\)[/tex].
