In the following arithmetic equation, correct the equality by interchanging the positions of exactly two operation signs: 4 – 10 × 5 + 9 ÷ 3 = 51. Which pair of signs must be interchanged to make the equation true under normal BODMAS rules?

Introduction / Context:
This question involves an arithmetic expression with multiple operations: subtraction, multiplication, addition and division. The expression is currently incorrect, and the task is to make it correct by interchanging exactly two operation signs. This tests understanding of both operator precedence rules, often remembered as BODMAS, and careful evaluation of expressions.

Given Data / Assumptions:

• The original equation is 4 – 10 × 5 + 9 ÷ 3 = 51.
• We must interchange two operation signs from among –, ×, + and ÷.
• Only one interchange is allowed, and the new expression must be evaluated using normal precedence rules: multiplication and division before addition and subtraction, and left to right at the same level.
• The right hand side is fixed at 51.
• We assume usual arithmetic on real numbers.

Concept / Approach:
An efficient approach is to compute the value of the original expression using BODMAS, then systematically test each possible swap suggested in the options. Each time, we recompute the value and check whether it equals 51. Care must be taken to apply precedence correctly. Because the expression is not very long, this trial and check method is straightforward and reliable.

Step-by-Step Solution:
Step 1: Evaluate the original expression 4 – 10 × 5 + 9 ÷ 3.Step 2: Apply multiplication and division first: 10 × 5 = 50 and 9 ÷ 3 = 3, so the expression becomes 4 – 50 + 3.Step 3: Now evaluate from left to right: 4 – 50 = -46, then -46 + 3 = -43. So the left hand side is -43, not 51.Step 4: Consider option a, swapping × and –. The new expression becomes 4 × 10 – 5 + 9 ÷ 3. Evaluate: 4 × 10 = 40, 9 ÷ 3 = 3, so 40 – 5 + 3 = 35 + 3 = 38, not 51.Step 5: Consider option b, swapping ÷ and ×. The new expression becomes 4 – 10 ÷ 5 + 9 × 3. Evaluate: 10 ÷ 5 = 2 and 9 × 3 = 27, so 4 – 2 + 27 = 2 + 27 = 29, not 51.Step 6: Consider option c, swapping + and –. The expression becomes 4 + 10 × 5 – 9 ÷ 3. Evaluate: 10 × 5 = 50 and 9 ÷ 3 = 3, so 4 + 50 – 3 = 54 – 3 = 51, which matches the right hand side.Step 7: Consider option d, swapping – and ÷. The expression becomes 4 ÷ 10 × 5 + 9 – 3. Evaluate: 4 ÷ 10 = 0.4, 0.4 × 5 = 2, so 2 + 9 – 3 = 11 – 3 = 8, not 51.Step 8: Only option c produces 51, so it is the correct choice.

Verification / Alternative check:
We can confirm the correctness of the modified expression for option c separately. With plus and minus swapped we have 4 + 10 × 5 – 9 ÷ 3. Using BODMAS, 10 × 5 = 50 and 9 ÷ 3 = 3. The expression becomes 4 + 50 – 3. Compute 4 + 50 = 54, and then 54 – 3 = 51. This equals the required right hand side, confirming that the equation becomes valid under this interchange.

Why Other Options Are Wrong:
For option a, the expression becomes 4 × 10 – 5 + 9 ÷ 3, which simplifies to 4 × 10 = 40 and 9 ÷ 3 = 3, giving 40 – 5 + 3 = 38, not 51. For option b, the expression becomes 4 – 10 ÷ 5 + 9 × 3, which leads to 4 – 2 + 27 = 29. For option d, the expression becomes 4 ÷ 10 × 5 + 9 – 3, which yields 8. None of these match 51, so they are incorrect.

Common Pitfalls:
A frequent mistake is to ignore operator precedence and evaluate operations strictly from left to right. This gives incorrect intermediate results and can lead to wrong conclusions about which swap works. Another pitfall is performing arithmetic errors in multiplication or division. Working carefully and applying BODMAS consistently is essential.

Final Answer:
The equation becomes correct when we interchange the signs + and –.