Two numbers are respectively 35% and 42% less than a third number. By what percentage is the second number less than the first number?

Introduction / Context: This question is about comparing two numbers that are both defined relative to a common base number. Each number is given as a certain percentage less than a third number. We are asked to find how much smaller the second number is compared with the first one, expressed as a percentage of the first. This type of problem is common in percentage comparison and relative change topics.

Given Data / Assumptions:

Let the third number be N. First number is 35% less than N. Second number is 42% less than N. We need the percentage by which the second number is less than the first number. All percentages are simple and based on the same third number N.

Concept / Approach: We express both numbers in terms of N, then compare them. If the first number is 35% less than N, it is 65% of N. If the second number is 42% less than N, it is 58% of N. Once both are expressed as multiples of N, we find the difference between them and express this difference as a percentage of the first number. The key formula is: percentage difference = (difference / reference) * 100 where the reference here is the first number.

Step-by-Step Solution: Step 1: Express the first number in terms of N. 35% less than N means it is 65% of N. First number = 65% of N = 0.65N. Step 2: Express the second number in terms of N. 42% less than N means it is 58% of N. Second number = 58% of N = 0.58N. Step 3: Find the absolute difference between the first and second numbers. Difference = 0.65N - 0.58N = 0.07N. Step 4: Express this difference as a percentage of the first number. Percentage by which second is less than first = (difference / first) * 100. = (0.07N / 0.65N) * 100. Step 5: Cancel N from numerator and denominator. = (0.07 / 0.65) * 100. Step 6: Compute 0.07 / 0.65. 0.07 / 0.65 ≈ 0.1076923. Step 7: Multiply by 100 to convert to percent. 0.1076923 * 100 ≈ 10.769%. Thus, the second number is approximately 10.769% less than the first number.

Verification / Alternative check: As a check, we can choose a convenient value for N, say 100. Then the first number is 65 and the second is 58. The difference is 65 - 58 = 7. Now compute the percentage of 7 relative to 65: (7 / 65) * 100 ≈ 10.769%. This matches the earlier result and confirms the calculation.

Why Other Options Are Wrong: Values like 11.23% or 10.4% are close but not equal to the exact fraction 7 / 65 when converted to percentage. 9.87% and 8.5% are clearly smaller than the correct value around 10.769%, and they correspond to different ratios than the one produced by 65 and 58.

Common Pitfalls: A common mistake is to subtract the percentages 42% and 35% directly and report 7% as the answer, without referencing the correct base. The 7% represents a difference relative to the third number N, not relative to the first number. Another error is to express the difference as a percentage of the second number instead of the first, which would give a slightly different value. Always pay close attention to which number is being used as the reference when expressing a percentage difference.

Final Answer: The second number is approximately 10.769% less than the first number.