Two numbers are 50 percent and 75 percent less than a third number. By what percentage must the smaller number be increased so that it becomes equal to the larger number?

Introduction / Context:
This question tests understanding of percentage decrease and percentage increase relative to a common reference value. Two numbers are described in relation to a third number, and you must determine by what percentage the smaller one must be increased to equal the larger one.

Given Data / Assumptions:
- Let the third reference number be T.
- The first number is 50 percent less than T.
- The second number is 75 percent less than T.
- We need the percentage increase required to raise the smaller number to the value of the larger number.

Concept / Approach:
If a quantity is 50 percent less than T, it is equal to 0.5T. If another is 75 percent less than T, it is 0.25T. To make the smaller number equal to the larger number, we increase 0.25T until it reaches 0.5T. The required percentage increase is based on the usual formula: percentage increase = (increase / original) * 100.

Step-by-Step Solution:
Step 1: Let the third number be T.Step 2: First number = T - 50% of T = 0.5T.Step 3: Second number = T - 75% of T = 0.25T.Step 4: Required increase to make second equal to first = 0.5T - 0.25T = 0.25T.Step 5: Percentage increase = (increase / original) * 100 = (0.25T / 0.25T) * 100 = 100%.Step 6: So the smaller number must be increased by 100 percent.

Verification / Alternative check:
Take a concrete example by assuming T = 100. Then the first number is 50 and the second is 25. To raise 25 to 50, we add 25. Using the percentage formula (25 / 25) * 100 gives 100 percent. Any other value of T will scale both numbers proportionally but the required percentage will remain 100 percent, so the result is consistent.

Why Other Options Are Wrong:
25 percent would only raise 25 to 31.25 in the example, which is far smaller than 50. A 50 percent increase would give 37.5, still less than 50. A 75 percent increase yields 43.75, which again does not equal 50. A 150 percent increase would make the second number larger than the first in general. Only a 100 percent increase makes the smaller quantity exactly equal to the larger.

Common Pitfalls:
Learners often confuse percent less with percent of a number and may take 50 percent and 75 percent as values of T rather than reductions from T. Another mistake is to compute the difference between 50 percent and 75 percent as 25 percent and wrongly claim this as the answer, forgetting that we are increasing the smaller number relative to its own value.

Final Answer:
The smaller number must be increased by 100% to become equal to the larger number.