In a certain code, the operation "#" between two numbers is defined so that 6 # 30 = 5, 8 # 24 = 3 and 7 # 28 = 4. Using the same pattern, what is the value of 8 # 40?

Introduction / Context:
This is a typical coded operation question where an artificial symbol "#" connects two numbers. A few examples are given, and your task is to uncover the rule that links the input pair to the output. These questions test your number sense and your ability to detect simple arithmetic patterns quickly and accurately.

Given Data / Assumptions:

• 6 # 30 = 5.
• 8 # 24 = 3.
• 7 # 28 = 4.
• We must find 8 # 40 following the same rule.
• The rule is assumed to be consistent for all examples.

Concept / Approach:
Look at the relationship between each pair of input numbers and the result. A natural first attempt is to check whether the result is a ratio like second number divided by first number, or some simple linear combination. Notice that in each example, the second number is divisible by the first, and the quotient looks promisingly equal to the result. This suggests that a # b might simply be b / a.

Step-by-Step Solution:
Step 1: For 6 # 30, compute 30 / 6 = 5, which matches the given result 5.Step 2: For 8 # 24, compute 24 / 8 = 3, which again matches the given result 3.Step 3: For 7 # 28, compute 28 / 7 = 4, consistent with the provided result 4.Step 4: The rule is therefore a # b = b / a.Step 5: Apply this to 8 # 40: we have 40 / 8 = 5.

Verification / Alternative check:
We should quickly check that there is no other simpler pattern that fits all three examples. If we tried a # b = a * b or a + b or a - b, they would not match the given results. Since b / a works perfectly in every case, it is the only simple consistent rule. Therefore 8 # 40 must equal 5, not any of the other distractor values.

Why Other Options Are Wrong:

• 4 and 3 correspond to quotients from other pairs and do not match 40 / 8.
• 8 and 10 could come from misreading or guessing a different arithmetic pattern, such as b / 5 or a + b / 4, which do not hold for all examples.

Common Pitfalls:
Sometimes students try overly complicated formulas involving addition and subtraction, or they use only one example to guess a pattern. Always verify your hypothesis against every given case. Here, the simple ratio b / a works cleanly and consistently, which is a strong indicator that it is the intended rule.

Final Answer:
Using the rule a # b = b / a, we get 8 # 40 = 5.