If the price of an article decreases from Rs 40 to Rs 30 and, as a result, the quantity demanded increases from an initial level of Q1 units to 7,500 units, and the point price elasticity of demand over this change is given as −1, what is the original quantity demanded Q1?

Introduction / Context:
This question tests the application of the point elasticity of demand formula to find an unknown original quantity demanded. Point elasticity relates small percentage changes in quantity demanded to small percentage changes in price at a specific point on the demand curve. It is very commonly tested in microeconomics questions involving demand analysis and pricing decisions.

Given Data / Assumptions:

• Initial price P1 = Rs 40.
• New price P2 = Rs 30.
• Initial quantity Q1 is unknown.
• New quantity Q2 = 7,500 units.
• Point price elasticity of demand (Ep) = −1.
• We use the usual point elasticity formula based on the initial price and quantity.

Concept / Approach:
The point elasticity of demand using the initial values is given by: Ep = (dQ / dP) * (P1 / Q1). In discrete form for small changes, we approximate: Ep ≈ (ΔQ / ΔP) * (P1 / Q1), where ΔQ = Q2 − Q1 and ΔP = P2 − P1. The sign of Ep is negative for a normal downward sloping demand curve because price and quantity move in opposite directions. In this question, Ep is given as −1, so we substitute and solve for the unknown Q1.

Step-by-Step Solution:
Step 1: Compute the change in quantity demanded. ΔQ = Q2 − Q1 = 7,500 − Q1. Step 2: Compute the change in price. ΔP = P2 − P1 = 30 − 40 = −10. Step 3: Write the point elasticity formula. Ep ≈ (ΔQ / ΔP) * (P1 / Q1). Step 4: Substitute Ep = −1, P1 = 40, ΔQ, and ΔP. −1 = [(7,500 − Q1) / (−10)] * (40 / Q1). Step 5: Simplify the expression. (7,500 − Q1) / (−10) = (Q1 − 7,500) / 10. So −1 = [(Q1 − 7,500) / 10] * (40 / Q1) = (4 * (Q1 − 7,500)) / Q1. Step 6: Multiply both sides by Q1 to solve for Q1. −Q1 = 4(Q1 − 7,500). −Q1 = 4Q1 − 30,000. Step 7: Bring like terms together. −Q1 − 4Q1 = −30,000, so −5Q1 = −30,000. Step 8: Divide both sides by −5 to get Q1. Q1 = 30,000 / 5 = 6,000 units.

Verification / Alternative check:
We can verify by plugging Q1 = 6,000 back into the elasticity formula. Then ΔQ = 7,500 − 6,000 = 1,500. ΔP = −10. So (ΔQ / ΔP) = 1,500 / (−10) = −150. P1 / Q1 = 40 / 6,000 = 1 / 150. Therefore Ep ≈ (−150) * (1 / 150) = −1, which matches the given elasticity value. This confirms that the original quantity demanded was 6,000 units.

Why Other Options Are Wrong:
Option A (9,000 units) would produce a different elasticity and would not give Ep equal to −1. Option B (4,500 units) and option C (10,500 units) similarly fail when substituted back into the elasticity formula. Only Q1 = 6,000 makes the computed elasticity exactly −1, which is required by the question. Thus, the other options do not satisfy the mathematical relationship.

Common Pitfalls:
Common mistakes include using the wrong price or quantity as the base in the elasticity formula, forgetting the negative sign in the change in price, or using an arc elasticity formula where the question expects point elasticity. Another error is doing algebraic manipulation incorrectly when solving for Q1. Writing each step clearly and checking the result by substitution can help avoid these errors.

Final Answer:
The original quantity demanded before the price change was 6000 units.