Two numbers are 30% and 37% less than a third number. By what percentage should the smaller number be increased so that it becomes equal to the larger number?

Introduction / Context:
This question tests comparative percentage change between two quantities that are both defined relative to a common reference number. It is a typical aptitude problem where one must express each number as a percentage of a base and then compute the required percentage increase from one to the other.

Given Data / Assumptions:

• Let the third number be T.
• First number is 30 percent less than T.
• Second number is 37 percent less than T.
• We are asked by what percent the smaller number must be increased to equal the larger number.
• We assume T is positive so that all percentages make sense.

Concept / Approach:
A number that is 30 percent less than T is equal to 70 percent of T. A number that is 37 percent less than T is equal to 63 percent of T. If we call the larger one A and the smaller one B, then: A = 0.70 * T, B = 0.63 * T. Required percentage increase from B to A is: [(A - B) / B] * 100.

Step-by-Step Solution:
Step 1: Let T be the base number. Step 2: First number A = 70 percent of T = 0.70 * T. Step 3: Second number B = 63 percent of T = 0.63 * T. Step 4: Clearly A > B, so B must be increased. Step 5: Required increase = A - B = (0.70T - 0.63T) = 0.07T. Step 6: Percentage increase relative to B = (0.07T / 0.63T) * 100. Step 7: Simplify 0.07T / 0.63T = 7 / 63 = 1 / 9. Step 8: 1 / 9 in percent form = (1 / 9) * 100 = 11.11 percent approximately.

Verification / Alternative check:
Choose a convenient value for T, for example T = 100. Then A = 70 and B = 63. Increase needed from 63 to 70 = 7. Percentage increase = 7 / 63 * 100 = 11.11 percent approximately. This numeric example confirms that the formula-based calculation is correct.

Why Other Options Are Wrong:
7 percent: Comes from confusing the absolute difference 7 with the percentage difference. 10 percent: A rough guess, but it does not match the true ratio 7 / 63. 18.92 percent: Much larger than the real value, often arising from incorrect denominator choice. 9 percent: Another common guess but not supported by the correct fraction 1 / 9.

Common Pitfalls:
A frequent mistake is to compute the percentage change using the larger number as the base instead of the smaller one. Another error is to confuse percentage less than with percentage of. Always convert verbal descriptions like "30 percent less" into precise multipliers such as 0.70, then handle the comparison with a proper fraction for percentage change.

Final Answer:
The smaller number must be increased by 11.11 percent to become equal to the larger number.