If the price of a book is reduced by ₹5, a buyer can purchase 2 more books for the same budget of ₹200. What was the original price per book?
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A₹ 25
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B₹ 30
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C₹ 20
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D₹ 15
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E₹ 35
Answer
Correct Answer: ₹ 25
Explanation
Introduction / Context:This is a linear relation between price and quantity purchased under a fixed budget. When the unit price decreases, the number of items bought increases. The relationship can be modeled with a simple rational equation and solved algebraically.
Given Data / Assumptions:
- Budget = ₹200.
- Original price per book = p (₹).
- Reduced price per book = p − 5 (₹).
- At the reduced price, the buyer gets 2 more books than before.
Concept / Approach:Number of books originally = 200 / p. Number after reduction = 200 / (p − 5). The statement “2 more” translates to 200/(p − 5) = 200/p + 2. Solve this for p to find an admissible positive solution.
Step-by-Step Solution:
200/(p − 5) = 200/p + 2200/(p − 5) − 200/p = 2200( p − (p − 5) ) / (p(p − 5)) = 2 ⇒ 200*5 / (p(p − 5)) = 21000 = 2p(p − 5) ⇒ p(p − 5) = 500p^2 − 5p − 500 = 0 ⇒ Discriminant = 25 + 2000 = 2025 ⇒ √2025 = 45p = (5 + 45)/2 = 25 (positive root; negative root discarded)Verification / Alternative check:Original price ₹25 ⇒ can buy 200/25 = 8 books. Reduced price ₹20 ⇒ can buy 200/20 = 10 books. Increase = 2 as required.
Why Other Options Are Wrong:
- ₹30, ₹20, ₹15, ₹35: Substitution contradicts the “2 more books” condition for a ₹200 budget.
Common Pitfalls:
- Mistakes in setting up the equation (e.g., swapping p and p − 5 in denominators).
- Accepting a negative price solution from the quadratic (which is inadmissible).
Final Answer:₹ 25