Verified Answer :

  • The problem requires dividing \(12\) by \(-3\).
  • Apply the integer division rule: a positive divided by a negative equals a negative.
  • Calculate the quotient's magnitude: \(12 \div 3 = 4\).
  • Combine the sign and magnitude to get \(-4\), which corresponds to option b.
  • The final answer is \(\boxed{-4}\).

Explanation

  1. Analyzing the Division Problem

The expression to evaluate is \(12 \div (-3)\). We are dividing a positive integer, \(12\), by a negative integer, \(-3\). To solve this, we recall the basic rules of division for integers:

  1. When dividing two numbers with different signs, the result is always negative.
  2. The absolute value of the quotient is found by dividing the absolute values of the dividend and the divisor.
  1. Performing the Calculation

We identify the dividend as \(12\) and the divisor as \(-3\).
Following the rules:

  1. Determine the sign: Since we are dividing a positive number by a negative number, the sign of the result will be negative.
  2. Calculate the magnitude: Divide the absolute value of the dividend by the absolute value of the divisor: \(|12| \div |-3| = 12 \div 3 = 4\).

Combining these steps, the quotient is \(-4\).

  1. Final Conclusion and Selecting the Option

We compare our result, \(-4\), with the given options:

A\(-3\)
B\(-4\)
C\(1171\)
D\(\sqrt{12}\)

The calculated result \(-4\) corresponds to option b. Thus, the final answer is \(\boxed{-4}\).

Examples

In finance, if a company incurs a total loss of \(\$12\) million spread equally over three quarters (represented as \(-3\) units of time), the loss per quarter can be represented by the expression \(\frac{-12}{3} = -4\). This calculation helps in understanding the rate of loss in a business or project, allowing managers to forecast future performance and adjust their strategies accordingly. Division of signed numbers is a fundamental tool for modeling trends that involve both gains and losses.