Two numbers are 35 percent and 50 percent less than a third number. By what percentage should the second number be increased so that it becomes equal to the first number?

Introduction / Context:
This question involves relative percentage decreases from a common reference and then asks for the percentage increase required to make the smaller number equal to the larger decreased number. It combines understanding of percentage decrease and percentage increase together.

Given Data / Assumptions:

• First number is 35 percent less than a third number.
• Second number is 50 percent less than the same third number.
• We must find by what percent the second number should be increased to equal the first number.

Concept / Approach:
Let the third number be 100 units for simplicity. Then the first and second numbers can be expressed as percentages of 100. After that, we compare the required increase from the second to the first and convert the difference into a percentage of the second number.

Step-by-Step Solution:
Let the third number = 100 units.First number = 35 percent less than 100 = 65 units.Second number = 50 percent less than 100 = 50 units.To make the second number equal to the first, it must be increased from 50 to 65.Increase needed = 65 - 50 = 15 units.Required percentage increase = (15 / 50) * 100.Compute 15 / 50 = 0.30.Therefore percentage increase = 0.30 * 100 = 30 percent.

Verification / Alternative check:
If the actual third number were different, say 200, then first number = 130 and second = 100. Increase needed from 100 to 130 is 30 units, which is still 30 percent of 100, confirming the result is independent of the chosen base value.

Why Other Options Are Wrong:
Values like 23.08 percent or 25 percent reflect miscalculations or comparisons with the wrong base, such as the third number instead of the second. The value 15 percent is simply the raw difference in percentage reductions (35 and 50) misinterpreted as the answer.

Common Pitfalls:
Many students subtract the percentage reductions directly, taking 50 minus 35 = 15 percent as the required increase. However, the base of the increase is the second number, not the third number. Always base the percentage on the quantity being changed.

Final Answer:
The second number must be increased by 30 percent to equal the first number.