The price of an article decreases from 100 rupees to 80 rupees, and quantity demanded increases from Q1 units to 4600 units. If the point elasticity of demand at the initial point is 0.75 in magnitude, what is the value of Q1?

Introduction / Context:
This question examines understanding of point elasticity of demand and how to apply elasticity formulas to solve for an unknown initial quantity demanded. The information provided includes an initial and final price, a final quantity, and the value of point elasticity at the initial point. The task is to use the relationship between percentage changes in quantity and price to find the unknown initial quantity.

Given Data / Assumptions:

• Initial price P1 is 100 rupees.
• New price P2 is 80 rupees.
• Initial quantity demanded is Q1 units (unknown).
• New quantity demanded Q2 is 4600 units.
• Point elasticity of demand at the initial point has magnitude 0.75.
• We interpret elasticity as percentage change in quantity divided by percentage change in price, using the initial values for approximation.

Concept / Approach:
Point elasticity of demand at the initial point is often approximated using the formula: elasticity (E) = (ΔQ / Q1) / (ΔP / P1), where ΔQ is the change in quantity and ΔP is the change in price. Here, we use the magnitude of elasticity, ignoring the negative sign that usually indicates an inverse relationship. We know E, P1, P2, and Q2 and must solve for Q1.

Step-by-Step Solution:
1. Compute the change in price: ΔP = P2 - P1 = 80 - 100 = -20.2. Compute the percentage change in price based on the initial price: (ΔP / P1) = -20 / 100 = -0.20.3. Let ΔQ = Q2 - Q1 = 4600 - Q1.4. The formula for the magnitude of elasticity is |E| = (ΔQ / Q1) / |ΔP / P1|.5. Given |E| = 0.75 and |ΔP / P1| = 0.20, we have 0.75 = (ΔQ / Q1) / 0.20.6. Multiply both sides by 0.20: ΔQ / Q1 = 0.75 * 0.20 = 0.15.7. So 4600 - Q1 = 0.15 Q1, which implies 4600 = 1.15 Q1.8. Solve for Q1: Q1 = 4600 / 1.15 = 4000 units.

Verification / Alternative check:
We can check by recomputing elasticity with Q1 equal to 4000. The change in quantity ΔQ is 4600 - 4000 = 600 units. Percentage change in quantity is 600 / 4000 = 0.15. Percentage change in price in magnitude is 20 / 100 = 0.20. Elasticity magnitude is 0.15 divided by 0.20, which equals 0.75, exactly matching the given value. This confirms that the initial quantity demanded must have been 4000 units.

Why Other Options Are Wrong:
Option A: 5000 units would yield a negative change in quantity, which is inconsistent with demand increasing when price falls in this example.
Option C: 3000 units would give a much larger percentage increase in quantity and therefore a larger elasticity value, not 0.75.
Option D: 2000 units would yield an even larger percentage increase in quantity, and elasticity would be far above the given figure.
Option E: 6000 units would imply that quantity fell in response to a price decrease, which contradicts normal downward sloping demand behaviour.

Common Pitfalls:
Learners may confuse point elasticity with arc elasticity and improperly use mid point formulas or plug values into the wrong places. Another common error is ignoring whether the percentage change is based on the initial or average value. Carefully writing the formula and clearly defining the initial values helps avoid these mistakes. It is also important not to lose track of signs; in magnitude based questions, working with absolute values simplifies the algebra while preserving the relationship.

Final Answer:
The initial quantity demanded Q1 is 4000 units.