To solve the multiplication of two mixed numbers [tex]\(3 \frac{2}{11}\)[/tex] and [tex]\(-3 \frac{3}{5}\)[/tex]:
1. Convert the mixed numbers into improper fractions:
- For [tex]\(3 \frac{2}{11}\)[/tex]:
[tex]\[
3 \frac{2}{11} = 3 + \frac{2}{11} = \frac{3 \cdot 11 + 2}{11} = \frac{33 + 2}{11} = \frac{35}{11}
\][/tex]
- For [tex]\(-3 \frac{3}{5}\)[/tex]:
[tex]\[
-3 \frac{3}{5} = -3 + \frac{-3}{5} = \frac{-3 \cdot 5 - 3}{5} = \frac{-15 - 3}{5} = \frac{-18}{5}
\][/tex]
2. Multiply the improper fractions:
[tex]\[
\frac{35}{11} \times \frac{-18}{5} = \frac{35 \cdot -18}{11 \cdot 5} = \frac{-630}{55}
\][/tex]
3. Simplify the resulting fraction if possible:
We can simplify [tex]\(\frac{-630}{55}\)[/tex] by dividing the numerator and the denominator by their greatest common divisor. The gcd of 630 and 55 is 5:
[tex]\[
\frac{-630 \div 5}{55 \div 5} = \frac{-126}{11}
\][/tex]
4. Convert the simplified improper fraction to a mixed number:
- Divide the numerator by the denominator to get the integer part and the remainder:
[tex]\[
\frac{-126}{11} = -12 \frac{6}{11}
\][/tex]
- Here, -126 divided by 11 is -12 with a remainder of 6. This remainder forms the fractional part:
[tex]\[
-12 \frac{6}{11}
\][/tex]
Thus, the product of [tex]\(3 \frac{2}{11}\)[/tex] and [tex]\(-3 \frac{3}{5}\)[/tex] is [tex]\( -12 \frac{6}{11} \)[/tex].
