If the price of an article decreases from Rs 25 to Rs 20 and the quantity demanded increases from Q1 units to 1500 units, and the point elasticity of demand is -1.25, what is the initial quantity demanded Q1?

Introduction / Context:
This question is another application of point elasticity of demand, but this time we are asked to find the initial quantity demanded rather than the initial price. When the price changes and elasticity is known, we can use the definition of point elasticity to work backwards and find the missing value. Such problems reinforce the practical use of elasticity formulas and algebraic manipulation in economic analysis and exam questions.

Given Data / Assumptions:
- Initial price P1 is 25 rupees.
- New price P2 is 20 rupees.
- Initial quantity Q1 is unknown.
- New quantity Q2 is 1500 units.
- Point elasticity of demand e is -1.25.
- We use the point elasticity formula around the initial point (P1, Q1).

Concept / Approach:
Point elasticity of demand can be written as e = (ΔQ / ΔP) * (P / Q), where P and Q refer to the initial price and quantity. Here ΔQ equals Q2 - Q1 and ΔP equals P2 - P1. We know e, P1, P2 and Q2 and must solve for Q1. Handling signs correctly is crucial because ΔP is negative when price falls and elasticity for a normal demand curve is negative.

Step-by-Step Solution:
Step 1: Compute the change in price: ΔP = P2 - P1 = 20 - 25 = -5. Step 2: Let ΔQ = Q2 - Q1 = 1500 - Q1. This will be positive if quantity rises. Step 3: Use the point elasticity formula: e = (ΔQ / ΔP) * (P1 / Q1). Step 4: Substitute known values: -1.25 = (1500 - Q1) / (-5) * (25 / Q1). Step 5: Simplify step by step. First, (1500 - Q1) / (-5) equals (Q1 - 1500) / 5. Step 6: The expression becomes -1.25 = (Q1 - 1500) / 5 * (25 / Q1). Step 7: Combine the constants: (25 / 5) = 5, so -1.25 = 5 * (Q1 - 1500) / Q1. Step 8: Multiply both sides by Q1 to clear the denominator: -1.25 * Q1 = 5 * (Q1 - 1500). Step 9: Expand the right side: -1.25 Q1 = 5Q1 - 7500. Step 10: Bring Q1 terms to one side: -1.25 Q1 - 5Q1 = -7500, so -6.25 Q1 = -7500. Step 11: Divide both sides by -6.25: Q1 = 7500 / 6.25 = 1200 units. Step 12: Thus the initial quantity demanded Q1 is 1200 units.

Verification / Alternative check:
Check the result by recalculating elasticity with Q1 = 1200. Then ΔQ = 1500 - 1200 = 300 and ΔP = -5. Use e = (ΔQ / ΔP) * (P1 / Q1) = (300 / -5) * (25 / 1200) = (-60) * (25 / 1200). Since 25 / 1200 equals 1 / 48, we have e = -60 * 1 / 48 = -60 / 48 = -1.25. This matches the given elasticity, confirming that 1200 units is correct.

Why Other Options Are Wrong:
900 units, 1800 units and 2000 units all produce different elasticity values if substituted back into the formula. None of them yield -1.25, so they cannot be the correct initial quantity demanded.

Common Pitfalls:
Students often mismanage the signs of ΔP and ΔQ or use the midpoint elasticity formula instead of the point elasticity expression. Algebraic mistakes when solving for Q1 are also frequent, such as not correctly distributing constants across parentheses. To avoid these issues, write the formula carefully, keep track of negative signs and always verify the answer by substituting it back into the original equation, as done above.

Final Answer:
The initial quantity demanded Q1 is 1200 units.