Two numbers are 10% and 20% less than a third number respectively. By what percentage must the second (smaller) number be increased to make it equal to the first number?

Introduction / Context:
This percentage problem compares two numbers that are each given as a percentage less than a third number. We are asked how much the smaller of the two must be increased (in percentage terms) to reach the larger one. It tests comfort with percentage reduction and percentage increase, which are not symmetric operations.

Given Data / Assumptions:
Let the third (original) number be T.The first number is 10% less than T.The second number is 20% less than T.We must find the percentage increase needed to raise the second number up to the first.

Concept / Approach:
Express each of the two numbers as a multiple of T. A number that is 10% less than T equals 0.9T; a number that is 20% less equals 0.8T. Then we calculate how much the smaller number (0.8T) must be increased to reach 0.9T. The percentage increase is calculated as (difference / original smaller number) × 100.

Step-by-Step Solution:
Step 1: First number (10% less than T) = T − 0.10T = 0.90T.Step 2: Second number (20% less than T) = T − 0.20T = 0.80T.Step 3: The increase required to go from the second number to the first is 0.90T − 0.80T = 0.10T.Step 4: The percentage increase is measured relative to the second number, which is 0.80T.Step 5: Percentage increase = (increase / original smaller number) × 100.Step 6: Compute this: percentage increase = (0.10T / 0.80T) × 100.Step 7: Simplify the fraction: 0.10T / 0.80T = 1 / 8 = 0.125.Step 8: Multiply by 100 to get 0.125 × 100 = 12.5%.

Verification / Alternative check:
Take a simple value for T, such as T = 100. Then the first number is 90 and the second number is 80. The increase needed is 90 − 80 = 10. The percentage increase from 80 to 90 is (10 / 80) × 100 = 12.5%. This numerical check confirms the algebraic result.

Why Other Options Are Wrong:
A 10% increase from 80 would give 88, not 90. A 15% increase would produce 92, overshooting the first number. An 8% increase would give only 86.4. None of these match the required target of 90, so they are incorrect.

Common Pitfalls:
One common error is to subtract the percentage differences directly (20% − 10% = 10%) and assume that the answer is 10%, forgetting that percentage increase should be calculated relative to the smaller number. Another mistake is to compute the increase relative to T instead of the second number itself.

Final Answer:
The second number must be increased by 12.5% to equal the first number.