Multiply. Simplify and write your answer as a mixed number.

[tex]\[ 2 \frac{1}{4} \cdot 3 \frac{2}{3} \][/tex]

A. [tex]\[ 8 \frac{1}{4} \][/tex]

B. [tex]\[ \frac{99}{12} \][/tex]

C. [tex]\[ 8 \frac{3}{12} \][/tex]

D. [tex]\[ 6 \frac{1}{6} \][/tex]



Answer :

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].