If the price of an article decreases from an initial price of Rs P1 to Rs 75 and, as a result, the quantity demanded increases from 1,000 units to 1,200 units, and the point elasticity of demand (in magnitude) over this change is 3.2, what is the original price P1?

Introduction / Context:
This question tests the application of the point price elasticity of demand formula to determine an unknown original price. The point elasticity concept links small percentage changes in quantity demanded to small percentage changes in price at a specific point on the demand curve. Here the price elasticity of demand in magnitude is given as 3.2, and we must work backwards to find the initial price before the change.

Given Data / Assumptions:

• Initial price P1 is unknown.
• New price P2 = Rs 75.
• Initial quantity Q1 = 1,000 units.
• New quantity Q2 = 1,200 units.
• Point price elasticity of demand Ep has magnitude 3.2 (that is, Ep ≈ −3.2 for a downward sloping demand curve).
• We use the standard point elasticity approximation based on initial price and quantity.

Concept / Approach:
Using the initial values, the point price elasticity of demand is approximated as: Ep ≈ (ΔQ / ΔP) * (P1 / Q1), where ΔQ = Q2 − Q1 and ΔP = P2 − P1. For a normal downward sloping demand curve, Ep is negative because price and quantity move in opposite directions. The question gives the magnitude 3.2, which we interpret as Ep = −3.2. By substituting known values and solving for P1, we can determine the original price that is consistent with the given elasticity.

Step-by-Step Solution:
Step 1: Compute the change in quantity demanded. ΔQ = Q2 − Q1 = 1,200 − 1,000 = 200 units. Step 2: Express the change in price in terms of P1. ΔP = P2 − P1 = 75 − P1. Step 3: Write the point elasticity formula. Ep ≈ (ΔQ / ΔP) * (P1 / Q1). Step 4: Substitute Ep = −3.2, ΔQ = 200, Q1 = 1,000. −3.2 = (200 / (75 − P1)) * (P1 / 1,000). Step 5: Simplify the expression. −3.2 = (200P1) / (1,000(75 − P1)) = (P1) / (5(75 − P1)). Step 6: Multiply both sides by 5(75 − P1). −3.2 * 5(75 − P1) = P1. −16(75 − P1) = P1. Step 7: Expand the left side. −1,200 + 16P1 = P1. Step 8: Group like terms. 16P1 − P1 = 1,200. 15P1 = 1,200. Step 9: Divide to find P1. P1 = 1,200 / 15 = Rs 80.

Verification / Alternative check:
To verify, substitute P1 = 80 into the elasticity formula. Then ΔP = 75 − 80 = −5. We already know ΔQ = 200 and Q1 = 1,000. Point elasticity Ep ≈ (ΔQ / ΔP) * (P1 / Q1) = (200 / −5) * (80 / 1,000) = (−40) * 0.08 = −3.2. This matches the given magnitude of 3.2, confirming that P1 = Rs 80 is correct.

Why Other Options Are Wrong:
Option A (Rs 85) would produce a different elasticity value when substituted back into the formula and would not yield magnitude 3.2. Option C (Rs 90) and option D (Rs 95) similarly fail to generate the correct elasticity when used as P1. Only Rs 80 produces the elasticity value of −3.2 that the question specifies.

Common Pitfalls:
Common errors include treating the elasticity as positive and ignoring the negative sign, miscomputing the change in price, or using the wrong base for P and Q in the formula. Algebraic mistakes when solving for P1 can also lead to incorrect answers. Writing each step clearly and checking the final elasticity by substitution can help confirm that the solution is consistent with the data.

Final Answer:
The original price P1 of the article was Rs 80.