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

Introduction / Context:
This question involves three numbers connected by two ratios and a total sum. It is an example of chained ratios, where we must convert the given pairwise ratios into a single combined set of proportional values for all three numbers, then use the total sum to find their actual values. The focus is on correctly linking the ratios and solving a simple linear equation.

Given Data / Assumptions:
• Let the three numbers be F (first), S (second) and T (third).
• F : S = 2 : 3.
• S : T = 5 : 8.
• F + S + T = 98.
• We must find the value of S, the second number.

Concept / Approach:
From F : S = 2 : 3, we can let F = 2k and S = 3k for some k. From S : T = 5 : 8, we can similarly let S = 5t and T = 8t for some t. By matching the two expressions for S, we can connect k and t, express all three numbers in terms of a single parameter and then solve for that parameter using the total sum 98. Finally, we compute S from that parameter.

Step-by-Step Solution:
Step 1: From F : S = 2 : 3, write F = 2k and S = 3k.Step 2: From S : T = 5 : 8, write S = 5t and T = 8t.Step 3: Since S is the same number in both expressions, set 3k = 5t.Step 4: Solve for k in terms of t: k = 5t / 3.Step 5: Substitute into F and S. Then F = 2k = 2 * (5t / 3) = 10t / 3; S = 3k = 3 * (5t / 3) = 5t.Step 6: T is already 8t. So F + S + T = (10t / 3) + 5t + 8t.Step 7: Combine terms: 5t + 8t = 13t. So sum = (10t / 3) + 13t = (10t + 39t) / 3 = 49t / 3.Step 8: Set 49t / 3 = 98. Multiply both sides by 3: 49t = 294, so t = 294 / 49 = 6.Step 9: The second number S = 5t = 5 * 6 = 30.

Verification / Alternative check:
With t = 6, we have S = 30, T = 8t = 48 and F = 10t / 3 = 10 * 6 / 3 = 20. Check the ratios: F : S = 20 : 30 = 2 : 3; S : T = 30 : 48 = 5 : 8. Check the sum: 20 + 30 + 48 = 98, which matches the given total. Hence all constraints are satisfied and S = 30 is consistent.

Why Other Options Are Wrong:
Values 10, 15 and 20 for S would break at least one of the given conditions. For example, if S = 20, then F from 2 : 3 would be 40/3, not an integer, and T from 5 : 8 would be 32, giving inconsistent totals. Only S = 30 leads to integer values for F and T and satisfies both ratio relationships and the overall sum.

Common Pitfalls:
Some students incorrectly try to add ratios directly or treat them as independent instead of linking them through the common term S. Another error is to set up inconsistent expressions for F, S and T that cannot be simultaneously satisfied. Always express ratios with a common variable where they intersect and then use the total sum equation to solve.

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