Two whole numbers are such that the ratio of the first number to the second number is 4:3 and the square of the first number exceeds the square of the second number by 112. What is the value of the smaller number?

Introduction / Context:
This question combines the concept of ratios with a condition involving squares of numbers. Problems involving squares often appear in quantitative aptitude because they require both algebraic understanding and careful arithmetic. Here, the ratio of two numbers is given, and a relationship between their squares allows us to find specific values for the numbers.

Given Data / Assumptions:

Two whole numbers are in the ratio 4 : 3 (first : second).
The square of the first number exceeds the square of the second number by 112.
Both numbers are whole numbers (non negative integers).
We are asked to find the smaller of the two numbers.

Concept / Approach:
If two numbers are in the ratio 4:3, we can represent them as 4k and 3k for some positive integer k. The condition about squares then becomes an equation involving k. Specifically, the difference of their squares is known. We can use the algebraic identity a^2 - b^2 = (a - b) * (a + b) to simplify calculations and solve for k. Once k is found, we can determine both numbers and identify the smaller one.

Step-by-Step Solution:
Let the two numbers be 4k and 3k, with 4k being the larger number. Then the square of the first number is (4k)^2 = 16k^2. The square of the second number is (3k)^2 = 9k^2. Given that the square of the first exceeds the square of the second by 112, so 16k^2 - 9k^2 = 112. Simplify the left side: 7k^2 = 112. Therefore k^2 = 112 / 7 = 16. So k = 4 (taking positive root because numbers are whole and positive). The two numbers are then 4k = 4 * 4 = 16 and 3k = 3 * 4 = 12. The smaller number is 12.

Verification / Alternative check:
Check the condition directly using the found numbers. The ratio 16:12 simplifies by dividing both numbers by 4 to 4:3, matching the given ratio. The squares are 16^2 = 256 and 12^2 = 144. Their difference is 256 - 144 = 112, which matches the problem statement exactly. This confirms that the calculations and algebraic setup are consistent.

Why Other Options Are Wrong:
Option 3 would produce numbers 4 * 3 = 12 and 3 * 3 = 9, whose squares differ by 144 - 81 = 63, not 112.
Option 4 would yield numbers 16 and 12 if taken as k, which confuses the role of k and the actual numbers and does not satisfy the square difference with the correct ratio simultaneously.
Option 36 is much larger and does not correspond to 3k when the other number is 4k under the given square difference condition.

Common Pitfalls:
Some learners set up the equation using the difference of the numbers instead of the difference of their squares, which leads to a different and incorrect equation.
Others forget to use the ratio and try to guess numbers directly, resulting in a time consuming and error prone approach.
Another common mistake is to treat 16k^2 - 9k^2 as (16 - 9)k instead of (16 - 9)k^2, which changes the algebra completely.

Final Answer:
The value of the smaller number is 12.