In a certain numerical code, the operation * does not represent ordinary multiplication. Instead, it is defined so that 70 * 10 = -7, 50 * 5 = -10 and 10 * 5 = -2. Using the same rule, what should be the value of 80 * 1 ?

Introduction / Context:
In this question, the symbol * has been redefined to represent a special operation rather than normal multiplication. We are given three examples that show how the operation behaves with different pairs of numbers and are asked to apply the same pattern to find 80 * 1. The presence of negative results for positive inputs suggests that the coded operation involves division combined with a sign change. This type of problem tests understanding of patterns in numeric operations and careful use of given examples.

Given Data / Assumptions:

• 70 * 10 is equal to -7.
• 50 * 5 is equal to -10.
• 10 * 5 is equal to -2.
• The same rule is consistently used for every pair of numbers.
• We need to compute 80 * 1 using this rule.

Concept / Approach:
Looking at the results, we notice that 70 divided by 10 is 7, 50 divided by 5 is 10 and 10 divided by 5 is 2. Each coded result matches the quotient but with a negative sign. This suggests that the operation * may be defined as the negative of the quotient a / b. Once we confirm that this rule fits all three examples, we can apply it to the new pair 80 and 1. This approach keeps the pattern simple and consistent, which is typical for exam reasoning questions.

Step-by-Step Solution:
Step 1: Examine the first example: 70 * 10 = -7. Compute 70 / 10 = 7. Step 2: The result of the coded operation is -7, which is just the negative of 7. So 70 * 10 appears to mean -(70 / 10). Step 3: Check the second example: 50 * 5 = -10. Compute 50 / 5 = 10 and then take the negative, giving -10. Step 4: Check the third example: 10 * 5 = -2. Compute 10 / 5 = 2 and then take the negative, giving -2. Step 5: Since all three examples follow the same rule, we conclude that a * b is defined as minus a divided by b. Step 6: Apply this definition to 80 * 1. Compute 80 / 1 = 80. Step 7: Take the negative of this quotient: -80. Step 8: Therefore, 80 * 1 equals -80 under the given coding rule.

Verification / Alternative check:
The pattern a * b = -(a / b) is extremely simple and fully explains the three given examples with no contradictions. There is no need to introduce more complicated formulas involving both numbers in other ways. Any alternative rule would need to reproduce exactly the same three results, which is unlikely if it is not equivalent to the negative quotient. Thus the rule and the final result of -80 for 80 * 1 are well justified.

Why Other Options Are Wrong:
Option 45 does not relate to 80 divided by 1 in any clear way and ignores the negative sign pattern. Option -25 might be incorrectly based on 100 divided by 4 or some unrelated calculation. Option 20 is the positive quotient but fails to include the negative sign present in all coded results. Option -8 corresponds to the quotient of 80 and 10 with a negative sign, which does not fit the given examples. Only -80 correctly follows the rule a * b = -(a / b).

Common Pitfalls:
A common error is to treat * as ordinary multiplication or to look for a linear combination such as a - b or a + b. Another mistake is to overlook division and not test the obvious relationship between 70, 10 and 7, or between 50, 5 and 10. Some test takers also assume that the result must be positive and discard negative patterns without checking. The best strategy is to systematically test division and other simple operations whenever clean integer quotients are visible in the data.

Final Answer:
With the coded rule a * b = minus a divided by b, the value of 80 * 1 is -80.