The difference between a number and two-fifth of that number is 510. What is 10% of that number?

Introduction / Context:
This question is a basic algebra and percentage problem based on a simple linear relationship. Instead of directly giving the number, the problem gives a relationship between the number and a fraction of it. Such questions are common in aptitude exams and help reinforce understanding of fractions, algebraic equations, and the calculation of a percentage of a quantity.

Given Data / Assumptions:

• The unknown number is not given directly.
• The difference between the number and two-fifth of that number is 510.
• We are asked to find 10% of that number.
• The unknown number is assumed to be a positive real number as in typical aptitude questions.

Concept / Approach:
The core idea is to represent the unknown number by a variable, say N, and then translate the English statement into an algebraic equation. The difference between N and (2/5) * N is equal to 510. Once we solve for N, we compute 10% of N, which is (10/100) * N. This step-by-step algebraic approach avoids guesswork and works well for any similar fraction-based relationship.

Step-by-Step Solution:
Step 1: Let the unknown number be N.Step 2: Two-fifth of the number is (2/5) * N.Step 3: The difference between the number and two-fifth of it is N - (2/5) * N.Step 4: According to the question, N - (2/5) * N = 510.Step 5: Simplify the left side: N - (2/5) * N = (3/5) * N.Step 6: So we have (3/5) * N = 510.Step 7: Solve for N: N = 510 * (5/3) = 510 * (5 ÷ 3).Step 8: 510 ÷ 3 = 170, so N = 170 * 5 = 850.Step 9: Now compute 10% of N: 10% of 850 = (10/100) * 850 = 85.

Verification / Alternative check:
To verify, plug N = 850 back into the original condition. Two-fifth of 850 is (2/5) * 850 = 2 * 170 = 340. The difference between 850 and 340 is 850 - 340 = 510, which matches the value given in the problem. Next, 10% of 850 is indeed 85. Since both the algebraic equation and the verification step confirm our computations, the result is reliable.

Why Other Options Are Wrong:
If we tried 75 as 10% of the number, the original number would be 750. The difference between 750 and two-fifth of 750 (which is 300) would be 450, not 510. For 95, the number would be 950, and the difference between 950 and 380 would be 570, still not 510. For 105, the number would be 1050, giving a difference of 630, which is also incorrect. Option 65 leads to a number of 650, producing a difference of 390. Thus none of these options satisfy the condition except 85.

Common Pitfalls:
One common mistake is to misinterpret "two-fifth" as 2/50 or to subtract the fraction incorrectly. Another mistake is to take 510 as 2/5 of the number rather than the difference between the number and 2/5 of it. Some learners also forget to multiply by the reciprocal when solving (3/5) * N = 510. Carefully converting the language into a correct algebraic equation and performing fraction operations step by step helps avoid these errors.

Final Answer:
10% of the number is 85.