In a certain puzzle, the symbol * represents a special operation on two numbers. It is given that 3 * 1 = 20, 6 * 1 = 50 and 2 * 6 = -40. Using the same rule consistently, what should be the value of 5 * 8 ?

Introduction / Context:
Here we are dealing with an artificial operation denoted by the symbol *. The question gives three examples of how this operation behaves and then asks us to extend that behaviour to a new pair of numbers. This is a typical number coding question that tests the ability to detect linear numeric patterns and to generalise them correctly. The presence of both positive and negative results suggests that subtraction is involved in some way, even though the symbol looks like ordinary multiplication.

Given Data / Assumptions:

• The coded operation * acts on two numbers a and b.
• 3 * 1 is defined as 20.
• 6 * 1 is defined as 50.
• 2 * 6 is defined as -40.
• The same single rule must be valid for all pairs of numbers.
• We are required to compute 5 * 8.

Concept / Approach:
Because the results are fairly large compared to the inputs and sometimes negative, we suspect a rule based on a difference multiplied by a constant. A simple and effective approach is to assume a linear pattern of the form a * b = k (a - b) where k is a constant to be determined. We can test this with the given examples. If all three examples can be explained by the same constant k, we will have a reliable rule for the operation and can easily evaluate 5 * 8.

Step-by-Step Solution:
Step 1: Consider 3 * 1 = 20. The difference a - b is 3 - 1 = 2. Step 2: If the operation is k times the difference, then k * 2 = 20, so k = 10. Step 3: Check this constant with 6 * 1. The difference is 6 - 1 = 5. Using k = 10, we get 10 * 5 = 50, which matches the given value. Step 4: Check with 2 * 6. Here the difference is 2 - 6 = -4. Multiplying by 10 gives 10 * (-4) = -40, again matching the given result. Step 5: Since all three examples work with a single rule, we conclude that a * b = 10 (a - b). Step 6: Now apply this rule to 5 and 8. Compute the difference: 5 - 8 = -3. Step 7: Multiply by 10 to get the coded value: 10 * (-3) = -30. Step 8: Therefore 5 * 8 equals -30 under the given coding rule.

Verification / Alternative check:
The fact that a single simple expression 10 (a - b) explains all three given examples is strong evidence that this is the intended pattern. Alternative attempts that mix addition, multiplication and arbitrary constants would usually fail to fit all three relations at the same time. The constant 10 is also natural here because the outputs look like rounded multiples of 10. This consistency across different input pairs confirms the reliability of the rule before we apply it to the new pair (5, 8).

Why Other Options Are Wrong:
Option 100 would correspond to 10 (5 + 5) and ignores the negative sign arising from 5 - 8. Option 30 neglects the fact that the difference is negative and treats 8 - 5 instead. Option -50 would require a difference of -5, which does not match the actual difference between 5 and 8. Option 70 has no connection with 10 (a - b) and seems to be a random distractor. Only -30 is consistent with the discovered rule that fits all three original examples.

Common Pitfalls:
Students often try to use the product a * b or the sum a + b directly and then force additional adjustments that only work for one example. Another error is to miss the importance of the sign and assume that all results must be positive, even though one of the examples clearly gives a negative number. This leads to incorrect rules such as 10 (a + b) minus a constant. Always pay attention to both magnitude and sign of the outputs when designing a rule, and confirm the pattern on every given pair before finalising it.

Final Answer:
Since the operation is defined by a * b = 10 (a - b), the value of 5 * 8 is -30.