Sure, I'd be happy to walk you through this step-by-step solution.
We need to multiply the mixed number [tex]\(2 \frac{3}{5}\)[/tex] by the fraction [tex]\(-\frac{4}{3}\)[/tex].
### Step 1: Convert the Mixed Number to an Improper Fraction
Mixed numbers are easier to work with when they are converted to improper fractions.
1.1 Start with the mixed number [tex]\(2 \frac{3}{5}\)[/tex].
1.2 First, convert the mixed number to an improper fraction.
- The integer part is [tex]\(2\)[/tex].
- The fractional part is [tex]\(\frac{3}{5}\)[/tex].
To convert [tex]\(2 \frac{3}{5}\)[/tex] to an improper fraction, you multiply the denominator [tex]\(5\)[/tex] by the integer part [tex]\(2\)[/tex] and add the numerator [tex]\(3\)[/tex]:
[tex]\[
2 \frac{3}{5} = \frac{2 \cdot 5 + 3}{5}
\][/tex]
[tex]\[
= \frac{10 + 3}{5}
\][/tex]
[tex]\[
= \frac{13}{5}
\][/tex]
### Step 2: Identify the Second Fraction
We need to multiply the improper fraction [tex]\(\frac{13}{5}\)[/tex] by [tex]\(-\frac{4}{3}\)[/tex]:
[tex]\[
\frac{13}{5} \cdot \left(-\frac{4}{3}\right)
\][/tex]
### Step 3: Multiply the Fractions
To multiply fractions, you simply multiply the numerators together and the denominators together:
[tex]\[
\frac{13}{5} \cdot \left(-\frac{4}{3}\right) = \frac{13 \cdot (-4)}{5 \cdot 3}
\][/tex]
[tex]\[
= \frac{-52}{15}
\][/tex]
### Step 4: Simplify the Result (if necessary)
In this case, the fraction [tex]\(\frac{-52}{15}\)[/tex] is already in its simplest form, so there is no need for further simplification.
### Conclusion
The product of [tex]\(2 \frac{3}{5}\)[/tex] and [tex]\(-\frac{4}{3}\)[/tex] is:
[tex]\[
\frac{-52}{15}
\][/tex]
Moreover, if we want to express the answer as a decimal:
[tex]\[
\frac{-52}{15} \approx -3.466666666666667
\][/tex]
So, the final answer in fraction form is [tex]\(\frac{-52}{15}\)[/tex] and in decimal form is approximately [tex]\(-3.466666666666667\)[/tex].
