Solve the expression −1/4 × { −45 − [ (−96) ÷ (−32) ] } using the correct order of operations and determine its exact numerical value.

Introduction / Context:
This question is a simplification problem involving fractions, negative numbers and nested brackets. It tests your ability to follow the correct order of operations (division before subtraction, and brackets first) and to handle negative signs carefully.

Given Data / Assumptions:

• Expression: −1/4 × { −45 − [ (−96) ÷ (−32) ] }.
• Standard arithmetic rules apply, with division done before addition or subtraction within the brackets.
• Multiplication by a negative fraction appears outside the brackets.

Concept / Approach:
We simplify from the inside out. First compute the inner division (−96) ÷ (−32), then handle the subtraction inside the curly braces, and finally multiply the result by −1/4. Being careful with signs at every step is crucial, because a negative multiplied by a negative yields a positive.

Step-by-Step Solution:
Step 1: Focus on the innermost bracket: (−96) ÷ (−32). Step 2: Division of two negative numbers gives a positive result: (−96) ÷ (−32) = 3. Step 3: Substitute this into the expression inside the curly braces: −45 − [ (−96) ÷ (−32) ] becomes −45 − 3. Step 4: Compute −45 − 3 = −48. Step 5: Now the full expression becomes −1/4 × { −48 }. Step 6: Multiplying a negative by a negative gives a positive result. Step 7: Compute the magnitude: 1/4 of 48 is 48 / 4 = 12. Step 8: Therefore, −1/4 × (−48) = +12. Step 9: The final value of the expression is 12.

Verification / Alternative check:
We can quickly check the calculation: the inner division 96 / 32 is clearly 3, and the signs cancel because both numerator and denominator are negative. Then we have −45 − 3 = −48. Multiplying −48 by −1/4 should yield a positive number with magnitude 48 / 4 = 12. No step violates the order of operations, so the result 12 is consistent.

Why Other Options Are Wrong:
10.5 and −10.5 would imply that the fraction applied was not 1/4 or that the subtraction inside the braces was done incorrectly. −12 would mean that the effect of one of the negative signs was ignored. 11 is simply a miscalculation of 48 / 4. Only 12 matches careful application of the arithmetic rules.

Common Pitfalls:
Students may forget that subtracting a positive 3 from −45 gives −48, or they might mishandle the negative signs when multiplying by −1/4. Another common mistake is to divide −96 by 32 instead of −32, giving the wrong sign in the intermediate step. Writing each step clearly and checking the sign at each operation helps avoid these errors.

Final Answer:
The exact value of the expression is 12.