Select the related number from the given alternatives: 5 : 124 :: 7 : ?

Introduction / Context:
This is a numerical analogy question that tests your ability to recognize a hidden pattern between two numbers and then extend that pattern to a new case. The pair 5 : 124 follows a specific mathematical relationship. You must find the same relationship between 7 and the unknown number. These questions commonly involve operations such as powers, addition, subtraction, or combinations of them.

Given Data / Assumptions:
- The first pair is 5 and 124, and the second pair is 7 and an unknown value.
- The relationship used to transform 5 into 124 is assumed to be arithmetic and consistent.
- The same relationship must be applied to 7 to get the correct number.
- Only one of the options 125, 248, 342, or 343 will satisfy the discovered rule.

Concept / Approach:
A good starting point for such analogies is to check for squares or cubes of the given number, because 124 is close to 125, which equals 5^3. That suggests that the pattern may involve a cube with a small adjustment like addition or subtraction. Once we identify that 124 is exactly one less than 125, we can write a formula and test it on the second number 7. The option that matches this formula is the answer.

Step-by-Step Solution:
Step 1: Compute 5^3. We get 5 * 5 * 5 = 125. Step 2: Compare this with the given result 124. We see that 124 = 125 - 1, so 124 = 5^3 - 1. Step 3: Formulate the rule as: for the number n, the related term is n^3 - 1. Step 4: Apply this rule to n = 7. Compute 7^3. We get 7 * 7 * 7 = 343. Step 5: Subtract 1 from 343. So 343 - 1 = 342. Step 6: The number related to 7 following the same pattern is therefore 342.

Verification / Alternative check:
To verify, we can check whether any alternative rule using simple operations could link 5 to 124 and still map 7 to some other option. For example, using n^3 alone would give 5^3 = 125, not 124, so that does not fit the first pair exactly. Any rule using squares or linear multiplication fails to produce 124 from 5 without arbitrary constants. The rule n^3 - 1 neatly fits the first pair and also yields one of the options for the second pair, which strongly validates it.

Why Other Options Are Wrong:
Option A "125" equals 5^3 or 5^3 for the first number, but it does not incorporate the subtraction of 1 used in the given pair, so it breaks the analogy pattern for 7.
Option B "248" has no direct connection to 7^3 or a simple variant of it, and does not match the structure n^3 - 1.
Option D "343" equals 7^3, but the first pair uses 5^3 - 1, not 5^3 itself. Therefore, choosing 343 would ignore the minus 1 adjustment present in the original pair.

Common Pitfalls:
Students often notice that 124 is close to 5^3 and jump to a cube based pattern but forget the small adjustment. Another common mistake is to apply inconsistent rules to the two pairs, like using n^3 in one and n^3 - 1 in the other. In a correct analogy, the same formula must apply consistently to both pairs.

Final Answer:
The correct related number is 342.