Question

Prep Exercise 6 When dividing two num bers that have the same sign, the result is ___. Prep Exercise 7 When dividing two numbers that have different signs, the result is ___. divide. $$\frac{-96}{8}$$

Short answer

Answer: The result of dividing -96 by 8 is -12. The rule dictating the sign of the result is that when dividing two numbers with different signs, the result is always negative. In this case, -96 (negative) and 8 (positive) have different signs, so the result is -12.

Step-by-step solution

Exercise 6: Dividing two numbers with the same sign

When dividing two numbers that have the same sign, the result is always positive. For example: $$\frac{10}{2} = 5$$ and $$\frac{-10}{-2} = 5$$

Exercise 7: Dividing two numbers with different signs

When dividing two numbers that have different signs, the result is always negative. For example: $$\frac{-10}{2} = -5$$ and $$\frac{10}{-2} = -5$$ Now, let's apply this understanding to the given division problem.

Applying the rule to solve our problem

Given the division problem, $$\frac{-96}{8}$$. In this case, the numerator is negative (-96) and the denominator is positive (8). Since the signs are different, the result will be negative, according to Exercise 7. To find the quotient, simply divide 96 by 8 and assign the appropriate sign: $$\frac{-96}{8} = -\frac{96}{8} = -12$$ Therefore, $$\frac{-96}{8} = -12$$.

Key concepts

Same Sign Division

Dividing two numbers that have the same sign, whether both are positive or both are negative, will always result in a positive quotient. Consider the scenario when both numbers are positive, such as dividing 10 by 2. Here, the answer is simply 5.

• Divide normally without considering the signs.
• The quotient remains positive.

A similar outcome occurs when both numbers are negative. For example, \( \frac{-10}{-2} = 5 \). Despite the presence of negative signs, the division of two negatives results once again in a positive quotient. This rule is consistent and helpful in ensuring accurate results during calculations.

Different Sign Division

When dividing two numbers with different signs, meaning one is positive and the other is negative, the quotient is always negative. For instance, dividing -10 by 2 gives us -5. This is because:

• Different signs mean the quotient takes a negative sign.
• Perform the division normally.

The pivotal concept here is recognizing the contrast between the signs. For example, when evaluating \( \frac{10}{-2} = -5 \), even though 10 divided by 2 equals 5, the negative sign from the denominator alters the overall sign of the quotient. This aligns with the rule that different signs equal a negative result.

Quotient Calculation

The heart of division problems, particularly involving integers, lies in accurately calculating the quotient. Consider the division problem \( \frac{-96}{8} \):First, determine the magnitude of the quotient by dividing the absolute values, thus 96 divided by 8 equals 12. Next:

• Assess sign influence. Here -96 (negative) and 8 (positive) imply a negative product.
• Apply the negative sign to 12, resulting in -12.

Hence, identifying the signs is crucial. Acquiring mastery over these scenarios fosters confidence in tackling any integer division problems, ensuring correct results every time.