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### Problem 6: Find the multiplicative inverse of [tex]\(\left(\frac{-2}{3} \div \frac{5}{21}\right)\)[/tex]
Step 1: Perform the division of the fractions.
To divide by a fraction, you multiply by its reciprocal. So:
[tex]\[ \frac{-2}{3} \div \frac{5}{21} = \frac{-2}{3} \times \frac{21}{5} \][/tex]
Step 2: Multiply the fractions.
Multiply the numerators and the denominators:
[tex]\[ \frac{-2 \times 21}{3 \times 5} = \frac{-42}{15} \][/tex]
Step 3: Simplify the fraction.
Simplify [tex]\(\frac{-42}{15}\)[/tex] by dividing both the numerator and the denominator by their greatest common divisor, which is 3:
[tex]\[ \frac{-42 \div 3}{15 \div 3} = \frac{-14}{5} \Rightarrow -2.8 \][/tex]
Step 4: Find the multiplicative inverse.
The multiplicative inverse of a number [tex]\(x\)[/tex] is [tex]\(\frac{1}{x}\)[/tex]. So for [tex]\(-2.8\)[/tex]:
[tex]\[ \text{Multiplicative Inverse} = \frac{1}{-2.8} \approx -0.35714285714285715 \][/tex]
Thus, the multiplicative inverse of [tex]\(\left(\frac{-2}{3} \div \frac{5}{21}\right)\)[/tex] is approximately [tex]\(-0.35714285714285715\)[/tex].
### Problem 7: Find the additive inverse of [tex]\(375 - (115 + 160)\)[/tex]
Step 1: Perform the operation inside the parentheses.
Add 115 and 160:
[tex]\[ 115 + 160 = 275 \][/tex]
Step 2: Subtract this result from 375.
[tex]\[ 375 - 275 = 100 \][/tex]
Step 3: Find the additive inverse.
The additive inverse of a number [tex]\(x\)[/tex] is [tex]\(-x\)[/tex]. So for 100:
[tex]\[ \text{Additive Inverse} = -100 \][/tex]
Thus, the additive inverse of [tex]\(375 - (115 + 160)\)[/tex] is [tex]\(-100\)[/tex].
### Problem 8: Find the value of [tex]\((-225) \times 550 + 1225 \times 550\)[/tex]
Step 1: Calculate each multiplication separately.
Calculate [tex]\((-225) \times 550\)[/tex]:
[tex]\[ -225 \times 550 = -123750 \][/tex]
Calculate [tex]\(1225 \times 550\)[/tex]:
[tex]\[ 1225 \times 550 = 673750 \][/tex]
Step 2: Add the results of the multiplications.
[tex]\[ -123750 + 673750 = 550000 \][/tex]
Thus, the value of [tex]\((-225) \times 550 + 1225 \times 550\)[/tex] is [tex]\(550000\)[/tex].
These are the detailed step-by-step solutions for the given problems.
### Problem 6: Find the multiplicative inverse of [tex]\(\left(\frac{-2}{3} \div \frac{5}{21}\right)\)[/tex]
Step 1: Perform the division of the fractions.
To divide by a fraction, you multiply by its reciprocal. So:
[tex]\[ \frac{-2}{3} \div \frac{5}{21} = \frac{-2}{3} \times \frac{21}{5} \][/tex]
Step 2: Multiply the fractions.
Multiply the numerators and the denominators:
[tex]\[ \frac{-2 \times 21}{3 \times 5} = \frac{-42}{15} \][/tex]
Step 3: Simplify the fraction.
Simplify [tex]\(\frac{-42}{15}\)[/tex] by dividing both the numerator and the denominator by their greatest common divisor, which is 3:
[tex]\[ \frac{-42 \div 3}{15 \div 3} = \frac{-14}{5} \Rightarrow -2.8 \][/tex]
Step 4: Find the multiplicative inverse.
The multiplicative inverse of a number [tex]\(x\)[/tex] is [tex]\(\frac{1}{x}\)[/tex]. So for [tex]\(-2.8\)[/tex]:
[tex]\[ \text{Multiplicative Inverse} = \frac{1}{-2.8} \approx -0.35714285714285715 \][/tex]
Thus, the multiplicative inverse of [tex]\(\left(\frac{-2}{3} \div \frac{5}{21}\right)\)[/tex] is approximately [tex]\(-0.35714285714285715\)[/tex].
### Problem 7: Find the additive inverse of [tex]\(375 - (115 + 160)\)[/tex]
Step 1: Perform the operation inside the parentheses.
Add 115 and 160:
[tex]\[ 115 + 160 = 275 \][/tex]
Step 2: Subtract this result from 375.
[tex]\[ 375 - 275 = 100 \][/tex]
Step 3: Find the additive inverse.
The additive inverse of a number [tex]\(x\)[/tex] is [tex]\(-x\)[/tex]. So for 100:
[tex]\[ \text{Additive Inverse} = -100 \][/tex]
Thus, the additive inverse of [tex]\(375 - (115 + 160)\)[/tex] is [tex]\(-100\)[/tex].
### Problem 8: Find the value of [tex]\((-225) \times 550 + 1225 \times 550\)[/tex]
Step 1: Calculate each multiplication separately.
Calculate [tex]\((-225) \times 550\)[/tex]:
[tex]\[ -225 \times 550 = -123750 \][/tex]
Calculate [tex]\(1225 \times 550\)[/tex]:
[tex]\[ 1225 \times 550 = 673750 \][/tex]
Step 2: Add the results of the multiplications.
[tex]\[ -123750 + 673750 = 550000 \][/tex]
Thus, the value of [tex]\((-225) \times 550 + 1225 \times 550\)[/tex] is [tex]\(550000\)[/tex].
These are the detailed step-by-step solutions for the given problems.
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