If 9 @ 7 = 4, 6 @ 1 = 10 and 7 @ 4 = 6 according to a certain rule, then what is the value of 8 @ 2 ?

Introduction / Context:
In this problem, the operator @ combines two numbers in a nonstandard way. Our job is to infer the rule of combination from three examples and then apply it to a new pair of numbers. The inputs are small, so we can experiment with simple arithmetic patterns such as sums, differences, products and multiples. This type of question assesses basic numerical pattern recognition.

Given Data / Assumptions:

• 9 @ 7 = 4.
• 6 @ 1 = 10.
• 7 @ 4 = 6.
• We must compute 8 @ 2 using the same rule.
• The operator @ is the same for all examples; there is one consistent definition.

Concept / Approach:
One of the simplest relationships to explore is the difference between the two numbers. For 9 and 7, the difference is 2, and the result 4 is exactly twice that difference. For 6 and 1, the difference is 5 and the result is 10, again twice the difference. If this pattern also holds for 7 and 4, then we can confidently apply it to 8 and 2. This kind of reasoning checks for a simple, consistent rule across all given pairs.

Step-by-Step Solution:
Step 1: Consider 9 @ 7. The difference 9 − 7 equals 2. Step 2: The given result is 4. Notice that 4 equals 2 × 2, or twice the difference. Step 3: Now check 6 @ 1. The difference 6 − 1 equals 5. Step 4: The given result is 10, and 10 equals 2 × 5, again twice the difference. Step 5: Check 7 @ 4. The difference 7 − 4 is 3. Step 6: The result is 6, which equals 2 × 3, confirming the same rule. Step 7: From these three cases we infer that a @ b = 2 × (a − b). Step 8: Apply this rule to 8 @ 2. The difference 8 − 2 equals 6. Step 9: Multiply by 2: 2 × 6 = 12. Step 10: Therefore 8 @ 2 equals 12.

Verification / Alternative check:
To verify, we double check each example with the formula. For 9 @ 7, 2 × (9 − 7) equals 4. For 6 @ 1, 2 × (6 − 1) equals 10. For 7 @ 4, 2 × (7 − 4) equals 6. All given outcomes fit perfectly. Since the same rule gives 12 for 8 @ 2, and 12 is one of the answer options, it is the unique correct choice.

Why Other Options Are Wrong:
Values 1, 26 and 35 do not correspond to twice the difference 8 − 2, which is 6. To get those values would require inconsistent or unrelated rules that do not fit all three given examples. Therefore, any option other than 12 contradicts the pattern established by the data.

Common Pitfalls:
Some candidates attempt to involve products or sums of the numbers before checking simple differences. Others might miscalculate the differences or forget to multiply by two. Always start with the most straightforward possibilities and confirm the pattern against every example provided before applying it to new numbers.

Final Answer:
Using the rule a @ b = 2 × (a − b), we find that 8 @ 2 equals 12.