Sum of three numbers with chained ratios: The sum of three numbers is 68. The ratio of the first to the second is 2 : 3, and the ratio of the second to the third is 5 : 3. Find the second number.

Introduction / Context:
This problem links three numbers using two ratios. Converting both to a single parameter lets you express all three numbers consistently and then use the total to solve for that parameter.

Given Data / Assumptions:

• First : Second = 2 : 3.
• Second : Third = 5 : 3.
• Sum = 68.

Concept / Approach:
Let First = 2a, Second = 3a. Also Second = 5b, Third = 3b. Equate Second values to connect a and b, then write all three in terms of b and use the sum to solve for b.

Step-by-Step Solution:
3a = 5b ⇒ a = 5b/3.First = 2a = 10b/3; Second = 5b; Third = 3b.Sum = 10b/3 + 5b + 3b = (10/3 + 8)b = 34b/3 = 68.b = 68 * 3 / 34 = 6; Second = 5b = 30.

Verification / Alternative check:
Numbers are 10b/3 = 20, 5b = 30, 3b = 18; total = 68 and ratios 2 : 3 and 5 : 3 hold.

Why Other Options Are Wrong:

• 20 and 48 are the first and twice the third respectively, not the second.
• 58 exceeds the total logic.

Common Pitfalls:

• Mismatching the “second” when equating across both ratios.

Final Answer:
30