- Solve for missing factors by dividing the product by the known factor, respecting sign rules.
- Compute missing products by multiplying the two factors, again paying attention to signs.
- For Problem 8: b. \(\boxed{-5}\), c. \(\boxed{8}\), d. \(\boxed{-147}\), e. \(\boxed{9}\), g. \(\boxed{-2}\), h. \(\boxed{-9}\).
- For Problem 9: a. \(\boxed{-3}\).
Explanation
- Data and Problem Analysis
The problem involves basic multiplication of integers and decimals, requiring us to solve for missing factors or products in given equations. Multiplication follows specific sign rules: the product of two numbers with the same sign is positive, while the product of two numbers with different signs is negative. We are also asked to evaluate a specific expression involving a fraction.
- Solving Equations (Problem 8)
To find missing factors in the equations, we rearrange the formula \(a \cdot b = c\) to solve for the missing variable:
- For (b), \(\_ \cdot 3 = -15\). Let the missing number be \(x\). Then \(x = -15 \div 3 = -5\).
- For (c), \(\_ \cdot 4 = 32\). Let the missing number be \(x\). Then \(x = 32 \div 4 = 8\).
- For (d), \(-49 \cdot 3 = \_\). This is a direct multiplication. \(-49 \times 3 = -147\).
- For (e), \((-2) \cdot (-4.5) = \_\). Multiplication of two negatives yields a positive. \(-2 \times -4.5 = 9\).
- For (g), \((-7) \cdot \_ = 14\). Let the missing number be \(x\). Then \(x = 14 \div (-7) = -2\).
- For (h), \(\_ \cdot (-10) = 90\). Let the missing number be \(x\). Then \(x = 90 \div (-10) = -9\).
- Evaluating Expression (Problem 9a)
To evaluate the expression \(\frac{1}{4} \cdot (-12)\), we multiply the fraction by the integer. Since the denominator is 4, we divide -12 by 4 and then multiply by the numerator 1: \(\frac{1 \cdot (-12)}{4} = \frac{-12}{4} = -3\).
- Final Results
The missing values for the equations are:
B\(-5\)
C\(8\)
D\(-147\)
E\(9\)
g. \(-2\)
h. \(-9\)
The value for the expression in 9a is \(-3\).
Examples
Understanding multiplication of signs is essential in fields like finance and science. For instance, in accounting, a positive value might represent revenue and a negative value might represent an expense. When calculating interest over time (which can be represented by negative growth) or managing multiple accounts with both profits and losses, correctly applying these multiplication rules is crucial for balancing books and tracking growth. Similarly, in physics, calculating force, work, or displacement often requires handling vectors with opposite directions, represented by positive and negative signs.