Select the related number from the given alternatives: 24 is to 60 as 210 is to which number?

Introduction / Context:
This is a numerical analogy that checks whether you can recognise patterns based on factorial like products of consecutive integers. The pair 24 : 60 is given, and you must work out how 210 should be completed in the same way. Such questions frequently use products of consecutive numbers or standard number patterns and are common in aptitude tests.

Given Data / Assumptions:
1) The first pair is 24 and 60.
2) The second pair gives 210 as the first number and asks for the related second number.
3) The options are 348, 336, 340, and 326.
4) We suspect that 24 and 60 may be products of consecutive natural numbers.
5) The same pattern must then be applied to 210 to find its partner number.

Concept / Approach:
Notice that 24 can be written as 4 * 3 * 2, which is the product of three consecutive integers. Similarly, 60 can be written as 5 * 4 * 3. This suggests a pattern where you start with three consecutive integers n, n minus 1, and n minus 2 to get the first number, and then move one step ahead to use n plus 1, n, and n minus 1 for the second number. The question is then to find n for which n * (n minus 1) * (n minus 2) equals 210 and apply the same idea.

Step-by-Step Solution:
Step 1: Express 24 in terms of products of consecutive integers. 24 equals 4 * 3 * 2. Step 2: Express 60 similarly. 60 equals 5 * 4 * 3. Step 3: Observe the pattern: first number uses 4, 3, 2, and second number uses 5, 4, 3. In other words, we have n * (n - 1) * (n - 2) for the first term and (n + 1) * n * (n - 1) for the second term when n equals 4. Step 4: For 210, find n such that n * (n - 1) * (n - 2) equals 210. Try n equals 7: 7 * 6 * 5 equals 210, which works perfectly. Step 5: Now compute the related number by using (n + 1) * n * (n - 1) with n equals 7. That gives 8 * 7 * 6. Step 6: Multiply these numbers. 8 * 7 equals 56, and 56 * 6 equals 336. Step 7: Check the options and see that 336 is present as option b.

Verification / Alternative check:
Verify that no other simple rule involving non consecutive products or fixed multipliers fits both 24 : 60 and 210 : 336 as neatly. For example, 24 multiplied by 2.5 gives 60, but 210 multiplied by 2.5 would give 525, which is not an option. The consecutive integer pattern elegantly covers both pairs, so it is the most reliable explanation.

Why Other Options Are Wrong:
The numbers 348, 340, and 326 cannot be obtained from 8 * 7 * 6 or from any consistent extension of the product pattern used for the first pair. Choosing any of these would imply changing the rule halfway through, which is not allowed in well formed analogies. Therefore, they must be treated as distractors only.

Common Pitfalls:
A common error is to look only for a fixed multiplier from the first number to the second, without checking for structure inside each number. Another mistake is to try random operations like adding digits or combining sums and products without any clear pattern. For product analogies, it is often helpful to factor the numbers into primes and see whether they can be grouped into simple sequences of consecutive integers as done here.

Final Answer:
Using the same consecutive integer product pattern, 210 corresponds to 336 in the analogy 24 : 60 :: 210 : ?.