The sum of three numbers is 162. The ratio of the first number to the second number is 5:7 and the ratio of the second number to the third number is 5:3. What is the value of the second number?

Introduction / Context:
This question combines two ratio relationships and a given total sum to find individual numbers. It is a common pattern in aptitude tests where you must merge separate ratios into a single combined ratio for all the quantities involved. Once that combined ratio is obtained, the actual values follow by simple proportional reasoning using the total sum.

Given Data / Assumptions:

There are three numbers whose sum is 162.
First : Second = 5 : 7.
Second : Third = 5 : 3.
All numbers are positive real numbers, typically treated as integers in such questions.
We need to find the second number specifically.

Concept / Approach:
To combine two ratios that share a common term, we express them with a common value for the shared term. Here the second number appears in both ratios, so we scale each ratio so that the second number has the same multiple in both. That gives us a unified ratio for first, second and third together. Then we use the total sum and the total number of parts in the ratio to find the value of one part and hence each number.

Step-by-Step Solution:
Given first : second = 5 : 7. Given second : third = 5 : 3. Let first = 5a and second = 7a from the first ratio. From the second ratio, let second = 5b and third = 3b. We need the expressions for the second number to match, so set 7a = 5b. Choose 7a = 5b = 35k, so a = 5k and b = 7k. Then first = 5a = 5 * 5k = 25k, second = 7a = 35k, third = 3b = 3 * 7k = 21k. Total sum = 25k + 35k + 21k = 81k = 162. Therefore k = 162 / 81 = 2. Second number = 35k = 35 * 2 = 70.

Verification / Alternative check:
With k = 2, the three numbers are 50, 70 and 42. Check the ratios: 50:70 simplifies to 5:7 and 70:42 simplifies to 5:3 when divided by 14. Also, the sum 50 + 70 + 42 = 162 matches the given total, confirming that the second number 70 is correct.

Why Other Options Are Wrong:
Option 35 is only half of the correct second number and would produce a sum much smaller than 162 if used in the ratio pattern derived above.
Option 80 does not fit the derived combined ratio, and plugging it in breaks the given relationships between first, second and third numbers.
Option 40 is also inconsistent with the required ratios and would not satisfy both ratio conditions simultaneously.

Common Pitfalls:
Students often add the ratios directly without first matching the common term, which leads to incorrect combined ratios.
Another mistake is to set 5:7 and 5:3 directly as 5:7:3 instead of correctly scaling them to align the shared term.
Some learners also forget to use the total sum of 162 to scale the combined ratio to actual numbers, leaving the answer only in parts, not in real values.

Final Answer:
The value of the second number is 70.