Two numbers are 30% and 60% less than a third number. By what percentage should the smaller number be increased so that it becomes equal to the larger number?

Introduction / Context:
This question tests relative percentage comparison between numbers defined in terms of a common reference. The two numbers are expressed as percentages of a third number, and we must find by how much the smaller one must be increased so that it equals the larger one. Such questions are typical in ratio and percentage topics in aptitude exams.

Given Data / Assumptions:
- There is a third number, call it T.
- First number A is 30 percent less than T.
- Second number B is 60 percent less than T.
- We must find the percentage increase needed for B to become equal to A.

Concept / Approach:
A number that is 30 percent less than T is equal to 70 percent of T. Similarly, one that is 60 percent less than T is 40 percent of T. Once A and B are expressed as fractions of T, we can compare them directly. The required percentage increase is the difference between A and B divided by B, multiplied by 100. It is important to consider B as the base, since we are increasing B.

Step-by-Step Solution:
Let the third number be T. A is 30% less than T, so A = T * (1 - 30 / 100) = 0.70T. B is 60% less than T, so B = T * (1 - 60 / 100) = 0.40T. We need to increase B so that it becomes equal to A. Required increase = A - B = 0.70T - 0.40T = 0.30T. Percentage increase based on B = (Required increase / B) * 100. So, percentage increase = (0.30T / 0.40T) * 100 = (0.75) * 100 = 75 percent.

Verification / Alternative check:
Take a simple value T = 100. Then A = 70 and B = 40. To raise B from 40 to 70, the increase needed is 30. As a percentage of B, this is (30 / 40) * 100 = 75 percent. The numerical method exactly matches our algebraic result, so the required percentage increase is confirmed as 75 percent.

Why Other Options Are Wrong:
42.86 percent and 50 percent: These percentages are too low; applying them to 40 does not give 70.
30 percent: This confuses the absolute difference 30 with the percentage increase, ignoring the correct base of 40.
60 percent: This is closer but still does not give the exact required value. Sixty percent of 40 is 24, which would give 64, not 70.

Common Pitfalls:
A typical mistake is taking the average of 30 and 60 or directly subtracting them, rather than correctly comparing the actual numbers A and B. Another pitfall is computing the percentage relative to the larger number instead of the smaller one. Remember, if we are increasing B, then B is the base for the percentage change formula.

Final Answer:
The smaller number must be increased by 75 percent to become equal to the larger number.