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Sagot :
To solve the problem of multiplying and simplifying the mixed numbers [tex]\(-7 \frac{1}{2}\)[/tex] and [tex]\(-2 \frac{1}{5}\)[/tex], follow these steps:
### Step 1: Convert Mixed Numbers to Improper Fractions
First, we need to convert the mixed numbers to improper fractions.
For [tex]\(-7 \frac{1}{2}\)[/tex]:
1. Multiply the whole number part by the denominator of the fractional part:
[tex]\(-7 \cdot 2 = -14\)[/tex]
2. Add the result to the numerator of the fractional part:
[tex]\(-14 + 1 = -15\)[/tex]
3. The improper fraction is:
[tex]\(-\frac{15}{2}\)[/tex]
For [tex]\(-2 \frac{1}{5}\)[/tex]:
1. Multiply the whole number part by the denominator of the fractional part:
[tex]\(-2 \cdot 5 = -10\)[/tex]
2. Add the result to the numerator of the fractional part:
[tex]\(-10 + 1 = -11\)[/tex]
3. The improper fraction is:
[tex]\(-\frac{11}{5}\)[/tex]
### Step 2: Multiply the Improper Fractions
To multiply the fractions, simply multiply the numerators and the denominators:
[tex]\[ \left( -\frac{15}{2} \right) \cdot \left( -\frac{11}{5} \right) = \frac{(-15) \cdot (-11)}{2 \cdot 5} = \frac{165}{10} \][/tex]
### Step 3: Simplify the Fraction
To simplify [tex]\(\frac{165}{10}\)[/tex]:
Divide both the numerator and the denominator by their greatest common divisor. In this case, the GCD of 165 and 10 is 5.
[tex]\[ \frac{165 \div 5}{10 \div 5} = \frac{33}{2} \][/tex]
### Step 4: Convert Improper Fraction to Mixed Number
To convert [tex]\(\frac{33}{2}\)[/tex] to a mixed number:
1. Divide the numerator by the denominator to get the whole number part:
[tex]\(33 \div 2 = 16\)[/tex] remainder [tex]\(1\)[/tex]
2. The whole number part is 16.
3. The fractional part is the remainder over the original denominator:
[tex]\(\frac{1}{2}\)[/tex]
Thus, the mixed number is:
[tex]\[ 16 \frac{1}{2} \][/tex]
### Conclusion
The result after multiplying and simplifying the given mixed numbers [tex]\(-7 \frac{1}{2}\)[/tex] and [tex]\(-2 \frac{1}{5}\)[/tex] is:
[tex]\[ \boxed{16 \frac{1}{2}} \][/tex]
Therefore, the correct answer is:
[tex]\( 16 \frac{1}{2} \)[/tex]
### Step 1: Convert Mixed Numbers to Improper Fractions
First, we need to convert the mixed numbers to improper fractions.
For [tex]\(-7 \frac{1}{2}\)[/tex]:
1. Multiply the whole number part by the denominator of the fractional part:
[tex]\(-7 \cdot 2 = -14\)[/tex]
2. Add the result to the numerator of the fractional part:
[tex]\(-14 + 1 = -15\)[/tex]
3. The improper fraction is:
[tex]\(-\frac{15}{2}\)[/tex]
For [tex]\(-2 \frac{1}{5}\)[/tex]:
1. Multiply the whole number part by the denominator of the fractional part:
[tex]\(-2 \cdot 5 = -10\)[/tex]
2. Add the result to the numerator of the fractional part:
[tex]\(-10 + 1 = -11\)[/tex]
3. The improper fraction is:
[tex]\(-\frac{11}{5}\)[/tex]
### Step 2: Multiply the Improper Fractions
To multiply the fractions, simply multiply the numerators and the denominators:
[tex]\[ \left( -\frac{15}{2} \right) \cdot \left( -\frac{11}{5} \right) = \frac{(-15) \cdot (-11)}{2 \cdot 5} = \frac{165}{10} \][/tex]
### Step 3: Simplify the Fraction
To simplify [tex]\(\frac{165}{10}\)[/tex]:
Divide both the numerator and the denominator by their greatest common divisor. In this case, the GCD of 165 and 10 is 5.
[tex]\[ \frac{165 \div 5}{10 \div 5} = \frac{33}{2} \][/tex]
### Step 4: Convert Improper Fraction to Mixed Number
To convert [tex]\(\frac{33}{2}\)[/tex] to a mixed number:
1. Divide the numerator by the denominator to get the whole number part:
[tex]\(33 \div 2 = 16\)[/tex] remainder [tex]\(1\)[/tex]
2. The whole number part is 16.
3. The fractional part is the remainder over the original denominator:
[tex]\(\frac{1}{2}\)[/tex]
Thus, the mixed number is:
[tex]\[ 16 \frac{1}{2} \][/tex]
### Conclusion
The result after multiplying and simplifying the given mixed numbers [tex]\(-7 \frac{1}{2}\)[/tex] and [tex]\(-2 \frac{1}{5}\)[/tex] is:
[tex]\[ \boxed{16 \frac{1}{2}} \][/tex]
Therefore, the correct answer is:
[tex]\( 16 \frac{1}{2} \)[/tex]
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