If the price of an article decreases from an initial price P1 to Rs 25, and as a result the quantity demanded increases from 900 units to 1200 units, and the (absolute) point elasticity of demand is 2, what is the initial price P1?

Introduction / Context:
Elasticity of demand is an important concept in economics that measures how responsive the quantity demanded of a good is to changes in its price. Point elasticity uses a specific formula based on small changes in price and quantity around a particular point. This question gives a change in price and quantity demanded, along with the value of point elasticity, and asks you to find the original price P1. It tests your ability to apply the point elasticity formula correctly and interpret percentage changes in quantity and price.

Given Data / Assumptions:

• Initial price is P1 (unknown), final price is 25 rupees.
• Initial quantity demanded is 900 units, final quantity demanded is 1200 units.
• Point elasticity of demand (in absolute value) is given as 2.
• We use the usual point elasticity formula based on the initial price and quantity.

Concept / Approach:
For small changes, the point price elasticity of demand (ignoring the minus sign for inverse relation) can be written as E = (ΔQ / ΔP) * (P1 / Q1), where ΔQ is the change in quantity demanded, ΔP is the change in price, P1 is the initial price, and Q1 is the initial quantity demanded. Here, ΔQ = 1200 - 900 = 300 units. The price decreases from P1 to 25, so ΔP = 25 - P1, which is negative, but we use the absolute value when equating to the given elasticity of 2. Solving this equation for P1 will give the required initial price.

Step-by-Step Solution:
Step 1: Compute the change in quantity: ΔQ = 1200 - 900 = 300 units.Step 2: Write the point elasticity formula in absolute value form: E = (ΔQ / ΔP) * (P1 / Q1).Step 3: Substitute E = 2, Q1 = 900, and ΔQ = 300. We use ΔP in magnitude as P1 - 25.Step 4: The equation becomes 2 = (300 / (P1 - 25)) * (P1 / 900).Step 5: Simplify: 2 = (300 * P1) / (900 * (P1 - 25)) = (P1) / (3 * (P1 - 25)).Step 6: Cross multiply: 2 * 3 * (P1 - 25) = P1, so 6 * (P1 - 25) = P1.Step 7: Expand: 6P1 - 150 = P1, so 5P1 = 150, giving P1 = 30 rupees.

Verification / Alternative check:
Check that this P1 value is consistent with the data. The price falls from 30 to 25, a change of 5 rupees. The quantity rises from 900 to 1200, a change of 300 units. Using the point elasticity formula again: E = (ΔQ / ΔP) * (P1 / Q1) in magnitude = (300 / 5) * (30 / 900) = 60 * (30 / 900) = 60 * (1 / 30) = 2. This matches the given elasticity of 2, confirming that P1 = 30 rupees is correct.

Why Other Options Are Wrong:
If P1 were 20 rupees, option A, the price would have increased to 25, which contradicts the statement that price decreases. P1 = 25, option C, would mean no change in price, making elasticity undefined in this context. P1 = 35, option D, would produce a different elasticity value; substituting into the formula would not give 2. Only P1 = 30 rupees yields the correct elasticity when combined with the given quantity changes.

Common Pitfalls:
Students sometimes confuse arc elasticity formulas with point elasticity formulas or forget to use the initial price and quantity in the fraction P1 / Q1. Another common error is mishandling the sign of ΔP and ΔQ, but since the question gives elasticity as a positive number, you can safely work with absolute values. To avoid mistakes, write down the formula carefully, substitute known values step by step, and simplify systematically, as shown above.

Final Answer:
Rs 30