Select the related number from the given alternatives: 25 is to 260 as 30 is to which number?

Introduction / Context:
This is a numerical analogy of the form 25 : 260 :: 30 : ?. The goal is to discover the arithmetic or algebraic rule that transforms 25 into 260 and then apply that rule to 30 in order to select the correct answer. Such questions often use simple patterns involving multiplication and addition that must be consistent for all pairs.

Given Data / Assumptions:

• The first pair given is 25 on the left and 260 on the right.
• The second starting number is 30, and we must determine its corresponding value.
• The options are 320, 310, 340, and 300.
• We assume a single arithmetic pattern is applied consistently to both 25 and 30.
• The pattern should be simple enough to be appropriate for a standard aptitude question.

Concept / Approach:
Since 260 ends with a zero, it suggests multiplication by a number that produces a multiple of 10. Notice that 260 can be written as 26 multiplied by 10, and 26 is 25 + 1. This leads to a compact rule: take the given number, add 1, and then multiply by 10. We can verify that this works for 25 and then apply the same idea to 30. If it matches one of the answer options, that is the intended pattern.

Step-by-Step Solution:
Step 1: Start with 25 in the first pair. Step 2: Add 1 to 25. This gives 25 + 1 = 26. Step 3: Multiply the result by 10. So 26 * 10 = 260, which matches the given second number in the pair. Step 4: Conclude that the working rule is (n + 1) * 10, where n is the given number. Step 5: Apply this rule to 30. First compute 30 + 1 = 31. Step 6: Multiply 31 by 10. So 31 * 10 = 310. Step 7: Compare with the answer options. 310 appears as option b, making it the correct choice.

Verification / Alternative check:
We can test whether any other simple rule fits 25 : 260. For instance, doubling 25 gives 50, and 25 squared is 625, neither of which equals 260. Writing 260 as 25 * 10 + 10 is just 10 times (25 + 1), which is another way to express the same rule (n + 1) * 10. This rule is straightforward and naturally explains 25 mapping to 260. Applying any alternative consistent rule to 30 would not reach a number present in the options as neatly as 310 does, confirming that our pattern is correct.

Why Other Options Are Wrong:
If we use the rule (n + 1) * 10 for 30, we get 310, not 320, 340, or 300. The values 320 and 340 would require adding 2 or 4 more tens, which does not match the relationship for 25. The number 300 would be 10 times 30, missing the plus one step that is clearly needed to obtain 260 from 25. Thus, these alternatives violate the consistent transformation required by the analogy.

Common Pitfalls:
A common mistake is to guess that 260 is simply 25 times some arbitrary factor and then try the same factor on 30, or to assume direct proportion without checking the structure of 260. Another pitfall is to overlook the simple 26 * 10 form of 260. In many exam questions, noticing that the right side can be factored into an incremented left side multiplied by a base such as 10 is the key to a quick solution.

Final Answer:
Using the consistent rule (number + 1) * 10, 25 maps to 260 and 30 maps to 310, so the correct answer is 310.