To solve the problem of multiplying [tex]\(4\)[/tex] by the mixed number [tex]\(-5 \frac{2}{3}\)[/tex], follow these detailed steps:
1. Convert the Mixed Number to an Improper Fraction:
- A mixed number consists of a whole part and a fractional part. Here, the mixed number is [tex]\(-5 \frac{2}{3}\)[/tex].
- Convert [tex]\(-5 \frac{2}{3}\)[/tex] into an improper fraction. To do this, multiply the whole number part by the denominator of the fractional part, then add the numerator. Don’t forget to consider the negative sign throughout the process.
- For [tex]\(-5 \frac{2}{3}\)[/tex]:
- The whole number part is [tex]\(-5\)[/tex].
- The fractional part is [tex]\(\frac{2}{3}\)[/tex].
- Multiply [tex]\(-5\)[/tex] by [tex]\(\frac{3}{3}\)[/tex], giving [tex]\(-15\)[/tex].
- Add [tex]\(\frac{2}{3}\)[/tex], resulting in [tex]\(-15 + \frac{2}{3} = -\frac{15}{3} + \frac{2}{3} = -\frac{17}{3}\)[/tex].
- So, [tex]\(-5 \frac{2}{3}\)[/tex] as an improper fraction is [tex]\(-\frac{17}{3}\)[/tex].
2. Multiply the improper fraction by 4:
- Now, multiply the improper fraction [tex]\(-\frac{17}{3}\)[/tex] by [tex]\(4\)[/tex].
- [tex]\[
4 \times \left(-\frac{17}{3}\right) = \left(4 \times -\frac{17}{3}\right)
\][/tex]
- This multiplication distributes as follows:
- The numerator: [tex]\(4 \times -17 = -68\)[/tex].
- The denominator remains [tex]\(3\)[/tex].
- So, [tex]\[
4 \left(-\frac{17}{3}\right) = -\frac{68}{3}
\][/tex]
3. Simplify the Fraction:
- The resulting fraction [tex]\(-\frac{68}{3}\)[/tex] is already in its simplest form, as the numerator and the denominator have no common factors other than 1.
Thus, the answer in its simplest form is:
[tex]\[
\boxed{-\frac{68}{3}}
\][/tex]
