A number is decreased to 72 1/2% (i.e., to 0.725 of itself) and the result is 87. Find the original number.
Correct Answer: 120
Introduction / Context: This is the mirror of “increase to” problems. “Decreased to 72 1/2%” means the final equals 72.5% of the original. We reverse the percent change to recover the initial value.
Given Data / Assumptions:
- Final value = 87.
- Final = 72 1/2% of original = 0.725 of original.
- 0.725 = 29/40 for exact calculation.
Concept / Approach: If original is x, then 0.725x = 87 ⇒ x = 87 / 0.725. Using fractions avoids rounding: x = 87 / (29/40) = 87 * 40 / 29.
Step-by-Step Solution:
Write 72 1/2% = 72.5% = 29/40.Let x be original ⇒ (29/40)x = 87.x = 87 * 40 / 29 = 3480 / 29 = 120.Verification / Alternative check: Check: 120 reduced to 72.5% ⇒ 120 * 0.725 = 87. Correct.
Why Other Options Are Wrong: 110, 135, 58 are results of using 27/40 or 3/4 mistakenly, or of subtracting percentages incorrectly.
Common Pitfalls: Interpreting “decreased by 72 1/2%.” The phrase here is “decreased to,” which means multiply by 0.725, not subtract 72.5 points.
Final Answer: 120