Fractional difference — If three-fourths of a number is subtracted from the number, the result is 163. Find the original number clearly and verify.

Introduction / Context:Linear equations with fractional coefficients appear frequently in aptitude tests. This problem states that removing three-fourths of a number from itself leaves 163. Recognizing that the remaining fraction is one-fourth leads to an immediate and accurate solution.

Given Data / Assumptions:

• Let the number be n.
• Statement: n - (3/4) * n = 163.
• All arithmetic is over real numbers; we expect a positive integer result.

Concept / Approach:Combine like terms on the left. Since n - (3/4) * n = (1/4) * n, the equation becomes a one-step calculation after isolating n by multiplying both sides by 4. This eliminates fractions and yields the answer directly. Always confirm with a quick substitution at the end.

Step-by-Step Solution:Start with n - (3/4) * n = 163.Compute left-hand side: (1/4) * n = 163.Multiply both sides by 4: n = 163 * 4 = 652.Therefore, the original number is 652.

Verification / Alternative check:Compute three-fourths of 652: (3/4) * 652 = 489. Then n - (3/4) * n = 652 - 489 = 163, exactly matching the condition.

Why Other Options Are Wrong:

• 625 / 562 / 632 / 544: Substituting any of these values into n - (3/4) * n does not yield 163; they produce different results.

Common Pitfalls:Misreading “three-fourths” as “three-fourth” of something else; forgetting that n - (3/4) * n equals (1/4) * n; arithmetic slips when multiplying 163 by 4.

Final Answer:652