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 to the third is 5 : 8. What is the value of the second number?

Introduction / Context:
This question involves combining two different ratios and a total sum to determine the individual values of three numbers. Such problems test a strong grasp of ratio chaining and the ability to create a common composite ratio. After generating a combined ratio for all three numbers, we compare it with the given total sum to obtain the actual values, particularly the second number in this case.

Given Data / Assumptions:
• Sum of three numbers = 98. • First : Second = 2 : 3. • Second : Third = 5 : 8. • All numbers are positive real numbers.

Concept / Approach:
We first express the three numbers in a single combined ratio. The second number appears in both ratios, so we make the second term common by using the least common multiple of its coefficients in the two separate ratios. After that, we add the ratio parts to match the total sum and then find the value of each number by multiplying the ratio parts by a common factor derived from 98.

Step-by-Step Solution:
Step 1: From First : Second = 2 : 3, write First = 2a and Second = 3a. Step 2: From Second : Third = 5 : 8, write Second = 5b and Third = 8b. Step 3: Since both represent the same second number, set 3a = 5b. Step 4: Choose a common value by taking a = 5k and b = 3k, so 3a = 15k and 5b = 15k. Step 5: Then First = 2a = 2 * 5k = 10k, Second = 3a = 15k, Third = 8b = 8 * 3k = 24k. Step 6: Combined ratio of the three numbers is 10 : 15 : 24. Step 7: Sum of ratio parts = 10 + 15 + 24 = 49. Step 8: Let the common multiplying factor be x, so total sum = 49x = 98 ⇒ x = 2. Step 9: Actual numbers are 10 * 2 = 20, 15 * 2 = 30, and 24 * 2 = 48. Step 10: The second number is therefore 30.

Verification / Alternative check:
We can verify by checking the stated ratios with the found numbers. First : Second = 20 : 30 = 2 : 3 and Second : Third = 30 : 48 = 5 : 8 after simplifying by dividing 30 and 48 by 6. Also, their sum 20 + 30 + 48 = 98 matches the given total. All conditions are satisfied.

Why Other Options Are Wrong:
• 10 and 20 are too small to maintain the total 98 under the given ratios. • 40 would make the sum of the three numbers exceed 98 under the required ratio structure.

Common Pitfalls:
A common error is to directly add or average the ratios without forming a common second term. Another mistake is to miscalculate the sum of ratio parts or the scaling factor. Skipping the verification step can cause candidates to overlook these small computational slips.

Final Answer:
Thus, the second number is 30.