Introduction / Context:
This question belongs to percentage comparison, a very important topic in quantitative aptitude. Instead of giving actual values, the problem describes two numbers in terms of their percentage reduction from a third number. We then need to find by what percentage the smaller number must be increased so that it becomes equal to the larger number. This tests your understanding of percentage decrease and percentage increase, and how they relate to each other with respect to a common base value.
Given Data / Assumptions:
- Let the third (reference) number be T.
- The first number is 10% less than T.
- The second number is 20% less than T.
- We need the percentage increase required on the second number so that it equals the first number.
- We assume all numbers are positive real numbers.
Concept / Approach:
We use the idea of expressing all numbers in terms of T. Then we compare the first and second numbers directly. If we call the first number A and the second number B, we are essentially asked: by what percentage should B be increased so that B becomes equal to A? The formula for percentage increase from B to A is:
percentage increase = [(A - B) / B] * 100%
Step-by-Step Solution:
Step 1: Express the first number A.
10% less than T means A = 90% of T = 0.9T.
Step 2: Express the second number B.
20% less than T means B = 80% of T = 0.8T.
Step 3: Find the difference A - B.
A - B = 0.9T - 0.8T = 0.1T.
Step 4: Compute the required percentage increase on B.
percentage increase = [(A - B) / B] * 100%.
= [0.1T / 0.8T] * 100%.
The T cancels out, giving (0.1 / 0.8) * 100%.
0.1 / 0.8 = 1 / 8 = 0.125.
So percentage increase = 0.125 * 100% = 12.5%.
Therefore, the second number must be increased by 12.5% to become equal to the first number.
Verification / Alternative check:
Pick a convenient value for T, such as T = 100. Then:
A = 10% less than 100 = 90.
B = 20% less than 100 = 80.
To go from 80 to 90, the increase is 10. The percentage increase based on 80 is:
(10 / 80) * 100% = (1 / 8) * 100% = 12.5%.
This numerical example confirms our algebraic solution.
Why Other Options Are Wrong:
25% would mean increasing 80 by 25% to 100, not to 90, so it is too large.
10% gives 80 + 10% of 80 = 80 + 8 = 88, which is less than 90.
15% gives 80 + 12 = 92, which is more than 90.
20% gives 80 + 16 = 96, which is also more than 90.
Only 12.5% makes the second number exactly equal to the first number.
Common Pitfalls:
One common mistake is to compute the percentage difference with respect to T instead of with respect to the second number B. Another frequent error is to subtract percentages directly (for example, taking 10% - 20% = -10%) without considering the base values. Students also sometimes confuse 12.5% with 12%, forgetting that 1 / 8 is exactly 12.5%. Writing everything symbolically in terms of T, then cancelling T, keeps the solution clear and avoids these mistakes.
Final Answer:
The second number must be increased by
12.5% to become equal to the first number.
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