Two numbers are 50% and 90% less than a certain third number.\nBy what percentage must the smaller of these two numbers be increased so that it becomes equal to the larger number?

Introduction / Context:
This question tests the ability to compare numbers that are given as percentage reductions from a common reference value. It then asks for the percentage increase required to move from the smaller number to the larger number. Such problems often appear in percentage and ratio sections of competitive examinations.

Given Data / Assumptions:

• A third reference number, call it T, is given implicitly.
• The first number (larger among the two derived numbers) is 50% less than T.
• The second number (smaller one) is 90% less than T.
• We must find the percentage increase required to raise the smaller number to the larger number.

Concept / Approach:
If a number is 50% less than T, then it equals T * (1 - 50/100) = 0.5 * T. If another number is 90% less than T, it equals T * (1 - 90/100) = 0.1 * T. To find how much the smaller must increase to become the larger, we take the difference between the two and divide by the smaller, then convert that ratio into a percentage. The formula is required percent increase = (larger - smaller) / smaller * 100.

Step-by-Step Solution:
Step 1: Let the reference number be T.Step 2: The larger number, A, is 50% less than T, so A = T * (1 - 50/100) = 0.5 * T.Step 3: The smaller number, B, is 90% less than T, so B = T * (1 - 90/100) = 0.1 * T.Step 4: The increase needed to go from B to A is A - B = 0.5 * T - 0.1 * T = 0.4 * T.Step 5: Required percentage increase = (A - B) / B * 100 = (0.4 * T / 0.1 * T) * 100.Step 6: Simplify the fraction: 0.4 * T / 0.1 * T = 4, so the required percentage = 4 * 100 = 400 percent.

Verification / Alternative check:
We can also choose a convenient value for T, such as T = 100. Then A = 50% less than 100 = 50. B = 90% less than 100 = 10. The increase required to go from 10 to 50 is 40. So the percentage increase = (40 / 10) * 100 = 400 percent. This numerical example confirms the algebraic reasoning and gives exactly the same result.

Why Other Options Are Wrong:
80 percent would be the required increase if the smaller number were 25, increasing to 45, which is not the case here.
40 percent would be correct if the smaller number were 100 and needed to move to 140, but that does not match the given relationships.
44.44 percent is a common trap value from inverse percentage problems but not applicable in this scenario.
60 percent is also another distractor that does not arise from the ratio between 50 and 10.

Common Pitfalls:
Students sometimes mistakenly subtract the percentage reductions (90% - 50% = 40%) and directly use that as the required increase, which is wrong because the increase must be measured relative to the smaller number, not relative to the reference number. Another pitfall is reversing the roles of the numerator and denominator when calculating the percent increase. Always remember that the percentage increase is based on the original smaller value as the reference. Using actual numbers after assigning a convenient value to T helps avoid these conceptual mistakes.

Final Answer:
The smaller number must be increased by 400 percent to become equal to the larger number.