Of three positive numbers, the ratio of the first to the second is 3 : 4 and the ratio of the second to the third is 5 : 6. The product of the second and third numbers is 4320. What is the sum of the three numbers?

Introduction / Context:
This question combines ratio relationships with a given product of two of the numbers. It is a typical aptitude problem where we are asked to find the sum of three numbers when multiple ratio links and a product condition are provided. It tests algebraic manipulation and understanding of how ratios translate into actual values using a common multiplier.

Given Data / Assumptions:

• First : Second = 3 : 4.
• Second : Third = 5 : 6.
• Product of Second and Third = 4320.
• The numbers are positive.
• We need the sum of all three numbers.

Concept / Approach:
We first express the three numbers in terms of a single parameter by combining the two given ratios. Once all three are expressed as multiples of one common variable, we will use the condition on the product of the second and third to solve for that variable. After we find the variable, we compute the individual numbers and then add them to obtain the sum.

Step-by-Step Solution:

Verification / Alternative check:
Check the product condition: second * third = 60 * 72 = 4320, which matches the given value. Check ratios: first : second = 45 : 60 = 3 : 4 and second : third = 60 : 72 = 5 : 6. Both conditions are satisfied, confirming correctness.

Why Other Options Are Wrong:

• 167, 187, and 197 do not correspond to any combination of three numbers that satisfy all the given ratio and product conditions.
• If we tried to adjust the numbers to reach those sums, at least one ratio or the product of the second and third would be violated.

Common Pitfalls:
Students may attempt to directly guess the numbers or may incorrectly combine ratios without resolving them to a single common variable. Another frequent error is to mis-handle the product equation and division operations, giving a wrong value for x. Carefully solving the algebra and checking the ratios at the end is a good safeguard.

Final Answer:
The sum of the three numbers is 177.