A custom operator "@" is defined so that 40 @ 8 = -5, 30 @ 3 = -10 and 20 @ 5 = -4. Using the same rule, what is the value of 60 @ 3 ?

Introduction / Context:
This is an example of an artificial or custom operator, where "@" represents some mathematical rule between two numbers. The task is not to guess randomly, but to infer the precise numerical relationship from the examples and then apply it to a new pair of numbers. Such questions test pattern recognition and basic arithmetic reasoning.

Given Data / Assumptions:

• 40 @ 8 = -5.
• 30 @ 3 = -10.
• 20 @ 5 = -4.
• We must find the value of 60 @ 3 using the same rule.
• The operator likely involves a simple combination such as division, multiplication or subtraction between the two numbers.

Concept / Approach:
To discover the rule, look for a simple calculation that converts the left pair into the result. Notice that in each case, the first number divided by the second gives a small positive integer, and the result shown is the negative of that quotient. Checking this pattern for all given examples helps confirm that the rule is consistent before applying it to the new pair.

Step-by-Step Solution:
Step 1: Evaluate 40 ÷ 8. This is 5. The given result is -5, which is the negative of the quotient. Step 2: Evaluate 30 ÷ 3. This is 10. The given result is -10, again the negative of the quotient. Step 3: Evaluate 20 ÷ 5. This is 4. The given result is -4, which continues the same pattern. Step 4: Conclude from all three examples that a @ b is defined as negative of a divided by b, or a @ b = - (a ÷ b). Step 5: Now apply this rule to 60 @ 3. Step 6: Compute 60 ÷ 3 = 20. Step 7: Take the negative of this quotient: -20. Step 8: Therefore, 60 @ 3 equals -20.

Verification / Alternative check:
We can verify by checking that no other simple pattern fits all three examples so neatly. For instance, subtraction 40 - 8 is 32, not related to -5 in a simple way. Similarly, ratios reversed as 8 ÷ 40 or products 40 × 8 do not produce the given outputs through a consistent simple rule. The negative quotient pattern uniquely matches each example exactly, which reinforces our conclusion.

Why Other Options Are Wrong:
Options 45 and 25 are positive numbers that would arise only if a candidate forgot to take the negative sign or used multiplication instead of division. Option -36 could come from miscomputing 60 ÷ 3 as 18 or from using some incorrect combination. Since the correct quotient is 20 and the rule requires a negative result, only -20 is valid.

Common Pitfalls:
A typical mistake is to apply a different rule to each example rather than seeking one rule that fits all of them. Another error is confusing the roles of the two numbers, for example dividing the smaller by the larger. Always test any guessed pattern on all given examples. If a proposed rule fails even once, it must be rejected.

Final Answer:
Using the rule a @ b = negative of a divided by b, the value of 60 @ 3 is -20.