The sum of three numbers is 67. The ratio of the first number to the second number is 3 : 5, and the ratio of the second number to the third number is 4 : 7. What is the value of the second number?

Introduction / Context:
This is a ratio and sum problem where we are asked to determine individual values of three numbers that are linked by ratios. Such questions evaluate comfort with manipulating ratios and converting them into actual numerical values using a common multiplying factor.

Given Data / Assumptions:

• Sum of three numbers = 67.
• First : Second = 3 : 5.
• Second : Third = 4 : 7.
• All three numbers are assumed to be positive real numbers.

Concept / Approach:
We have two ratio relations, each involving the second number. We convert them into a single ratio involving all three numbers by finding a common representation for the second number. After that, we know the total sum, so we can solve for the common multiplying factor and hence each individual number.

Step-by-Step Solution:
Let first : second = 3 : 5, so first = 3k and second = 5k.Let second : third = 4 : 7, so second = 4m and third = 7m.Equate the second number: 5k = 4m.Take k = 4t and m = 5t so that second = 20t.Then first = 3k = 12t and third = 7m = 35t.Sum of numbers = 12t + 20t + 35t = 67t.Given sum is 67, so 67t = 67, hence t = 1.Therefore, the second number = 20t = 20.

Verification / Alternative check:
Numbers are 12, 20 and 35. Check the ratios: 12 : 20 simplifies to 3 : 5 and 20 : 35 simplifies to 4 : 7. Their sum equals 12 + 20 + 35 = 67, which matches the given total. Thus the second number is correctly found as 20.

Why Other Options Are Wrong:
Values 24, 18 and 16 do not satisfy the stated ratios when combined with any corresponding first and third numbers while keeping the sum equal to 67. Only 20 works consistently with both ratio conditions and the total sum constraint.

Common Pitfalls:
Learners sometimes mix up the order of ratios or try to add ratios directly. Another pitfall is forgetting to match the middle term of both ratios by using a common multiple. Without that step, the derived numbers will not satisfy both ratio relationships simultaneously.

Final Answer:
The value of the second number is 20.