Verified Answer :

  • Identify that the quotient of two negative numbers is positive: \(\frac{-54}{-6} = \frac{54}{6}\).
  • Divide the absolute values: \(54 \div 6 = 9\).
  • Conclude that the result is \(\boxed{9}\).

Explanation

  1. Problem Analysis

The problem asks to evaluate the quotient of two negative integers: \(-54\) divided by \(-6\). The expression is written as \(\frac{-54}{-6}\). To solve this, we need to apply the rules of division for integers, specifically focusing on signs.

  1. Rule of Signs for Division

When dividing two integers, the sign of the quotient is determined by the following rules:

  • The quotient of two numbers with the same sign is positive.
  • The quotient of two numbers with different signs is negative.

In this case, both the numerator (\(-54\)) and the denominator (\(-6\)) are negative. Therefore, the result of the division must be a positive number. Specifically, \(\frac{-54}{-6} = \frac{54}{6}\).

  1. Performing the Division

Now we perform the division of the absolute values: \(54 \div 6\). We look for an integer \(n\) such that \(6 \times n = 54\). We know that \(6 \times 9 = 54\). Thus, \(54 \div 6 = 9\). Since the result is positive, the quotient is 9.

  1. Conclusion

Combining these steps, we have \(\frac{-54}{-6} = 9\). The final result is \(\boxed{9}\).

Examples

In everyday situations, dividing negative numbers can arise when dealing with net changes over time. For example, if a company loses \(\$54\) in total over several days of equal loss, and we know they lost \(\$6\) each day, we can calculate the number of days they were losing money by dividing the total loss by the daily loss: \(\frac{-54}{-6} = 9\). This tells us that the loss occurred over 9 days. Understanding the sign rules helps in correctly interpreting such financial or physical data.

Related Questions