Two numbers are 90% and 75% less than a certain third number. By what percentage should the smaller number be increased so that it becomes equal to the larger number?

Introduction / Context:
This question combines relative percentage comparisons with percentage increase. It involves three numbers, where two are described as being 90% and 75% less than a third number. The task is to find how much the smaller number should be increased, in percentage terms, so that it equals the larger number. It is a common pattern in ratio and percentage problems.

Given Data / Assumptions:

• Let the third number be N.
• First number is 90% less than N.
• Second number is 75% less than N.
• We must find the required percentage increase of the smaller number to reach the larger number.

Concept / Approach:
If a number is 90% less than N, it means it is 10% of N. Similarly, 75% less than N means 25% of N. We convert these verbal descriptions into actual values using N as a base. Then we consider the smaller number as the base value from which we need to reach the larger number, and compute the percentage increase as (difference / smaller) * 100%. The final answer will be independent of N, so we can leave N symbolic.

Step-by-Step Solution:
Step 1: Let the third number be N. Step 2: First number is 90% less than N, so it is (100% - 90%) of N = 10% of N = 0.10N. Step 3: Second number is 75% less than N, so it is (100% - 75%) of N = 25% of N = 0.25N. Step 4: The smaller number is 0.10N and the larger number is 0.25N. Step 5: Required increase to go from 0.10N to 0.25N is 0.25N - 0.10N = 0.15N. Step 6: Percentage increase = (increase / original smaller number) * 100% = (0.15N / 0.10N) * 100%. Step 7: Simplify the ratio 0.15N / 0.10N = 1.5. Step 8: Therefore, percentage increase = 1.5 * 100% = 150%.

Verification / Alternative check:
Choose a convenient value for N, say N = 100. Then the smaller number = 10% of 100 = 10, and the larger number = 25% of 100 = 25. To go from 10 to 25, the increase is 15. Percentage increase = (15 / 10) * 100% = 150%. This numerical check confirms the algebraic result.

Why Other Options Are Wrong:

• 250: This would mean the new number is 3.5 times the old one, which is not the case here.
• 200: This would imply the larger number is three times the smaller, but from 10 to 25 the ratio is 2.5.
• 100: This would be correct only if the larger number were exactly double the smaller number.
• 300: This is too high and indicates a confusion between absolute difference and relative difference.

Common Pitfalls:
Students sometimes confuse “x percent less than N” with “x percent of N” and may treat 90% less as 90% of N. Another common error is to compute percentage difference based on the larger number instead of the smaller one when calculating percentage increase. Always identify clearly which value is being used as the base for the percentage calculation.

Final Answer:
The smaller number must be increased by 150% to become equal to the larger number.