Sure, let's solve the problem step-by-step.
We are given the division of two mixed numbers:
[tex]\[ -3 \frac{2}{3} \div 2 \frac{1}{5} \][/tex]
First, let's convert these mixed numbers into improper fractions:
### Step 1: Convert mixed numbers to improper fractions
1. Convert [tex]\(-3 \frac{2}{3}\)[/tex] to an improper fraction:
- Multiply the whole number by the denominator: [tex]\((-3) \times 3 = -9\)[/tex].
- Add the numerator: [tex]\(-9 + 2 = -11\)[/tex].
- So, [tex]\(-3 \frac{2}{3} = -\frac{11}{3}\)[/tex].
2. Convert [tex]\(2 \frac{1}{5}\)[/tex] to an improper fraction:
- Multiply the whole number by the denominator: [tex]\(2 \times 5 = 10\)[/tex].
- Add the numerator: [tex]\(10 + 1 = 11\)[/tex].
- So, [tex]\(2 \frac{1}{5} = \frac{11}{5}\)[/tex].
### Step 2: Division of fractions
To divide fractions, we will multiply by the reciprocal of the second fraction:
[tex]\[ -\frac{11}{3} \div \frac{11}{5} = -\frac{11}{3} \times \frac{5}{11} \][/tex]
### Step 3: Simplify the multiplication
Next, we multiply the fractions:
[tex]\[ -\frac{11}{3} \times \frac{5}{11} \][/tex]
Multiply the numerators:
[tex]\[ -11 \times 5 = -55 \][/tex]
Multiply the denominators:
[tex]\[ 3 \times 11 = 33 \][/tex]
So, we have:
[tex]\[ -\frac{55}{33} \][/tex]
### Step 4: Simplify the fraction
Simplify [tex]\(-\frac{55}{33}\)[/tex]:
- Both the numerator and denominator can be divided by their greatest common divisor, which is 11:
[tex]\[ -\frac{55 \div 11}{33 \div 11} = -\frac{5}{3} \][/tex]
### Step 5: Convert to a mixed number
To convert [tex]\(-\frac{5}{3}\)[/tex] to a mixed number:
- Divide the numerator by the denominator: [tex]\(5 \div 3 = 1\)[/tex] remainder [tex]\(2\)[/tex].
- So, [tex]\(-\frac{5}{3} = -1 \frac{2}{3}\)[/tex].
Hence, the final result is:
[tex]\[ -3 \frac{2}{3} \div 2 \frac{1}{5} = -1 \frac{2}{3} \][/tex]
Among the given choices:
- [tex]\(-\frac{3}{5}\)[/tex]
- [tex]\(-1 \frac{2}{3}\)[/tex]
- [tex]\(\frac{3}{5}\)[/tex]
- [tex]\(1 \frac{2}{3}\)[/tex]
The correct answer is:
[tex]\[ -1 \frac{2}{3} \][/tex]
