Reverse-engineering an original fraction: After increasing the numerator of a fraction by 200% and the denominator by 150%, the resulting fraction is 9/35. What was the original fraction?
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A3/14
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B3/10
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C2/15
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D2/7
Answer
Correct Answer: 3/14
Explanation
Introduction / Context:Percentage increases on the numerator and denominator scale a fraction multiplicatively. “Increased by 200%” means the new numerator equals old numerator × 3. “Increased by 150%” means the new denominator equals old denominator × 2.5. You are given the final fraction and must recover the original one by reversing these multiplicative effects.
Given Data / Assumptions:
- Let the original fraction be n/d.
- New numerator after +200%: 3n.
- New denominator after +150%: 2.5d.
- Resulting fraction = 9/35.
Concept / Approach:Set 3n / (2.5d) = 9/35. Solve for the ratio n/d by isolating it on one side. The operations are pure scaling; there is no need to assume integer numerators or denominators at intermediate steps. Finally, simplify the result to lowest terms to match one of the options.
Step-by-Step Solution:
3n / (2.5d) = 9/35.Rearrange: n/d = (9/35) × (2.5/3).Compute: 2.5/3 = 5/6; hence n/d = (9/35) × (5/6) = 45/210 = 3/14.Verification / Alternative check:
Apply the stated increases to 3/14: new numerator = 3 × 3 = 9; new denominator = 2.5 × 14 = 35. Final = 9/35, which matches, so the original is correct.Why Other Options Are Wrong:
- 3/10, 2/15, 2/7: After applying the same percentage increases, these do not yield 9/35; numerical checks disagree.
Common Pitfalls:
- Misreading “increased by 200%” as “becomes 200%” (×2) instead of ×3.
- Forgetting that +150% equals ×2.5, not ×1.5 of the denominator.
Final Answer:
3/14