In the following pattern, the usual arithmetic symbols do not keep their ordinary meanings:\n\nIf 25 + 5 ÷ 2 = 40 and 35 + 5 ÷ 2 = 60, then using the same hidden rule, what should be the value of 45 + 5 ÷ 2?

Introduction / Context:
This is a number analogy question in which the symbols do not represent normal operations. Instead, the entire expression 25 + 5 ÷ 2 produces 40 according to some hidden rule, and 35 + 5 ÷ 2 produces 60. We must discover that consistent rule and then apply it to 45 + 5 ÷ 2 to obtain the missing value.

Given Data / Assumptions:

• Expression 1: 25 + 5 ÷ 2 results in 40.
• Expression 2: 35 + 5 ÷ 2 results in 60.
• The small numbers 5 and 2 are the same in all expressions.
• The same rule must apply to the case 45 + 5 ÷ 2.

Concept / Approach:
Rather than treating + and ÷ as normal operations, we look at how the first number is transformed into the result. If we compare 25 and 40, and 35 and 60, both results are close to twice the first number. This suggests a rule of the form result = 2 × (first number) – constant. Once we identify the constant using the two given examples, we can apply the same pattern to the third case.

Step-by-Step Solution:
Step 1: From 25 to 40, the increase is 40 - 25 = 15. Step 2: From 35 to 60, the increase is 60 - 35 = 25. Step 3: Check whether the expression 2 × first number - 10 fits both results. Step 4: For 25: 2 × 25 - 10 = 50 - 10 = 40, which matches the given value. Step 5: For 35: 2 × 35 - 10 = 70 - 10 = 60, again matching the given value. Step 6: Apply the same rule to 45: 2 × 45 - 10 = 90 - 10 = 80. Step 7: Therefore, 45 + 5 ÷ 2 should give 80 according to the established pattern.

Verification / Alternative check:
We might try other simple forms like result = first number + some multiple of 5, but these do not consistently fit both initial expressions. The linear rule 2 × first number - 10 is the simplest pattern that works perfectly for both 25 and 35 and so is the most reasonable hidden rule.

Why Other Options Are Wrong:
If we choose 70, 60, or 90, none of these follow from the rule 2 × first number - 10. For 45, this rule uniquely gives 80. Other answers would require inventing a different rule that breaks the consistency across all three expressions, so they must be rejected.

Common Pitfalls:
One common mistake is to assume that the + and ÷ symbols must be used directly and start doing mixed arithmetic with 25, 5 and 2. Another error is to look only at the differences 15 and 25 without seeing the underlying doubling pattern. Focusing on how the first number alone transforms into the result usually makes such puzzles easier to decode.

Final Answer:
Using the consistent rule result = 2 × first number - 10, we get 45 + 5 ÷ 2 = 80, so option (d) is correct.