The sum of three numbers is 98; the ratio of the first number to the second is 2 : 3 and the ratio of the second number to the third is 5 : 8; find the value of the second number.

Introduction / Context:
This question is a typical example of using ratios and sums to determine individual values. When multiple ratios between the same set of numbers are given, they can be combined into a single set of proportional parts. Then the total sum is used to scale these parts to actual numbers. Such problems appear frequently in quantitative aptitude and arithmetic reasoning sections.

Given Data / Assumptions:
- There are three numbers, say A, B, and C.
- Their sum is A + B + C = 98.
- The ratio of the first to the second, A : B, is 2 : 3.
- The ratio of the second to the third, B : C, is 5 : 8.
- All numbers are positive real numbers and are assumed to be consistent with the given ratios.

Concept / Approach:
We have two separate ratios involving the same variable B. To combine them, we express A and C in terms of B and introduce a common scaling factor. Alternatively, we set A, B, and C as multiples of some base quantity using the ratio chain. Once we obtain a complete combined ratio for A : B : C, we compare the sum of these parts with the given total 98 and find the value of one part. From there, it is straightforward to compute the required number, which is the value of B.

Step-by-Step Solution:
Step 1: Let A : B = 2 : 3. This means A = 2k and B = 3k for some positive constant k. Step 2: Let B : C = 5 : 8. This means B = 5m and C = 8m for some positive constant m. Step 3: Because B appears in both ratios, we equate the expressions for B: 3k = 5m. Step 4: Solve for k and m in terms of a new constant t. Take k = 5t and m = 3t so that 3k = 15t and 5m = 15t, keeping B consistent. Step 5: Substitute back to express each number. Then A = 2k = 2 * 5t = 10t, B = 3k = 15t, and C = 8m = 8 * 3t = 24t. Step 6: Add the three numbers: A + B + C = 10t + 15t + 24t = 49t. Step 7: Use the given sum 98 to write 49t = 98, which gives t = 2. Step 8: Substitute t = 2 to find B: B = 15t = 15 * 2 = 30.

Verification / Alternative check:
When t = 2, the numbers are A = 10 * 2 = 20, B = 30, and C = 24 * 2 = 48. Their sum is 20 + 30 + 48 = 98, which matches the given total. The ratio A : B is 20 : 30, which simplifies to 2 : 3, and the ratio B : C is 30 : 48, which simplifies to 5 : 8. Both ratios are satisfied. Therefore, B = 30 is consistent with all pieces of given information.

Why Other Options Are Wrong:
Option 27: If B were 27, the combined ratio and sum conditions would fail; we cannot create A and C that both honour the ratios and add up to 98.
Option 23: This value is too small and does not satisfy the derived proportional structure when we try to reconstruct A and C.
Option 33: This value is too large in the combined ratio framework and does not yield a correct total sum of 98.
Option 35: Similar to 33, this number does not satisfy both the ratio and sum constraints simultaneously.

Common Pitfalls:
One frequent error is to combine ratios incorrectly, for example by adding them directly instead of using a common multiple approach. Another mistake is to forget that the same B must satisfy both A : B and B : C simultaneously, which requires consistent scaling. Some learners also attempt guesswork with options instead of setting up the ratios algebraically, which can work occasionally but is unreliable for more complex problems.

Final Answer:
The value of the second number is 30.