Solve the expression 80 ÷ [48 − 56 − (60 − 36 ÷ 12 × 4)] by carefully applying the correct order of operations (BODMAS or PEMDAS) and find its exact value in simplest fractional form.

Introduction / Context:
This problem is designed to test your understanding of the order of operations in arithmetic expressions involving multiple nested brackets. You must correctly apply the rules for brackets, division and multiplication before subtraction, and then perform the final division to express the answer as a simple fraction.

Given Data / Assumptions:

• Expression: 80 ÷ [48 − 56 − (60 − 36 ÷ 12 × 4)].
• Standard order of operations applies (brackets first, then division and multiplication from left to right, followed by addition and subtraction).
• The final answer should be a simplified fraction.

Concept / Approach:
We evaluate from the innermost bracket outward. Inside the parentheses (60 − 36 ÷ 12 × 4), we must first perform 36 ÷ 12 and then multiply by 4, before subtracting from 60. The result is then used inside the square bracket [48 − 56 − (result)], which we simplify step by step. Finally, we divide 80 by the simplified bracket value.

Step-by-Step Solution:
Step 1: Consider the inner parentheses: (60 − 36 ÷ 12 × 4). Step 2: Perform the division: 36 ÷ 12 = 3. Step 3: Then multiply by 4: 3 × 4 = 12. Step 4: Now the inner parentheses become: 60 − 12 = 48. Step 5: Substitute this result into the square bracket: [48 − 56 − (60 − 36 ÷ 12 × 4)] becomes [48 − 56 − 48]. Step 6: Compute from left to right: 48 − 56 = −8. Step 7: Then −8 − 48 = −56. Step 8: So the full expression reduces to 80 ÷ (−56). Step 9: Compute the division: 80 ÷ (−56) = 80 / (−56) = −80 / 56. Step 10: Simplify −80 / 56 by dividing numerator and denominator by 8: (−80 ÷ 8) / (56 ÷ 8) = −10 / 7. Step 11: Therefore, the value of the expression is −10 / 7.

Verification / Alternative check:
We can verify by checking the intermediate bracket value. If the bracket equals −56, then 80 divided by −56 should give a negative fraction with magnitude 80 / 56. Simplifying 80 / 56 by the greatest common divisor 8 indeed gives 10 / 7, so the magnitude is correct and the sign is negative because we divided by a negative quantity.

Why Other Options Are Wrong:
−5 / 7 and −20 / 7 have the wrong magnitude, indicating mistakes in simplifying 80 / 56 or miscomputing the bracket. The positive fractions 5 / 7 and 10 / 7 ignore the negative sign from the bracket value, which is a sign error. Only −10 / 7 matches the correct step-by-step evaluation.

Common Pitfalls:
Frequent errors include performing 36 ÷ 12 × 4 as 36 ÷ (12 × 4) instead of handling division and multiplication from left to right, and miscalculating the bracket expression 48 − 56 − 48. Another common issue is forgetting to simplify the final fraction or dropping the negative sign when dividing by a negative number.

Final Answer:
The exact value of the expression is −10 / 7.