Two numbers are 40 percent and 80 percent less than a third number. By what percentage must the smaller of these two numbers be increased so that it becomes equal to the larger number?

Introduction / Context:
This is a percentage comparison problem that checks understanding of percentage decrease and percentage increase. The numbers are expressed as being a certain percent less than a reference number, and then we must find how much to increase one to match the other. Such relative percentage questions are very common in quantitative aptitude tests, especially in topics like profit and loss, ratios, and data interpretation.

Given Data / Assumptions:

• There is a third number, call it T.
• First number is 40 percent less than T.
• Second number is 80 percent less than T.
• We need the percentage increase required to raise the smaller number so that it equals the larger number.

Concept / Approach:
We express both numbers in terms of T. Forty percent less than T means 60 percent of T, and eighty percent less than T means 20 percent of T. The smaller number is clearly the one that is 20 percent of T. The percentage increase required is calculated by (difference between target and original) divided by the original, multiplied by 100. Using T keeps the algebra simple and shows that the final answer does not depend on the actual value of T.

Step-by-Step Solution:
Let the third number be T.First number is 40 percent less than T, so it equals 60 percent of T.So first number = 0.6T.Second number is 80 percent less than T, so it equals 20 percent of T.So second number = 0.2T.The smaller number is 0.2T and the larger is 0.6T.We want to find the percentage increase required to raise 0.2T to 0.6T.Increase required = 0.6T - 0.2T = 0.4T.Percentage increase = (increase / original) * 100 = (0.4T / 0.2T) * 100.Simplify 0.4T / 0.2T = 2.So percentage increase = 2 * 100 = 200 percent.

Verification / Alternative check:
Take a simple numeric example such as T = 100.Then first number = 60 and second number = 20.To change 20 into 60, the increase is 60 - 20 = 40.Percentage increase = (40 / 20) * 100 = 200 percent.This matches the algebraic result, confirming the answer.

Why Other Options Are Wrong:
100 percent would mean doubling the smaller number, taking it from 20 to 40, which is still below 60.33.3 percent and 66.6 percent are typical distractors related to ratios like 1/3 and 2/3, but they do not convert 0.2T to 0.6T.Only 200 percent increases the smaller number exactly to the larger number.

Common Pitfalls:
Some learners mix up percent less and percent of, which can lead to incorrect expressions for the two numbers.Another mistake is to compute the percentage change relative to the larger number instead of the smaller original number.It is also common to forget that 80 percent less than T means only 20 percent of T remains, not 80 percent of T.

Final Answer:
The smaller number must be increased by 200 percent to become equal to the larger number.