If the price of an article decreases from Rs P1 to Rs 190 and the quantity demanded increases from 5000 units to 5200 units, and if the point elasticity of demand is -0.8, what is the initial price P1?

Introduction / Context:
This numerical question tests the application of the point elasticity of demand formula. Instead of directly giving prices and asking for elasticity, it gives elasticity and some changes in price and quantity and asks for the initial price. Such problems help students understand the relationship between percentage changes in quantity and price and the elasticity measure, which is very important for pricing decisions, tax analysis and competitive exam preparation in Indian economy.

Given Data / Assumptions:
- Initial price is P1 (unknown).
- New price is 190 rupees.
- Initial quantity demanded is 5000 units.
- New quantity demanded is 5200 units.
- Point elasticity of demand is -0.8.
- Use the point elasticity formula based on initial price and initial quantity.

Concept / Approach:
For small changes, the point elasticity of demand can be approximated by the formula: elasticity (e) = (ΔQ / ΔP) * (P / Q), where ΔQ is the change in quantity, ΔP is the change in price, P is the initial price and Q is the initial quantity. Here elasticity and both quantities are known, and we must solve for P1. The sign of elasticity is negative for normal downward sloping demand curves, so we must handle the sign of ΔP carefully when substituting values.

Step-by-Step Solution:
Step 1: Compute the change in quantity: ΔQ = Q2 - Q1 = 5200 - 5000 = 200 units. Step 2: Express the change in price: ΔP = P2 - P1 = 190 - P1. Since we know only P2 and not P1, we keep this as an algebraic expression. Step 3: Write the point elasticity formula using initial values: e = (ΔQ / ΔP) * (P1 / Q1). Step 4: Substitute known values: -0.8 = (200 / (190 - P1)) * (P1 / 5000). Step 5: Simplify the fraction 200 / 5000 to 2 / 50 or 1 / 25. The equation becomes -0.8 = (1 / 25) * (P1 / (190 - P1)). Step 6: Multiply both sides by 25 to clear the denominator: -20 = P1 / (190 - P1). Step 7: Multiply both sides by (190 - P1): -20 * (190 - P1) = P1. Step 8: Expand the left side: -3800 + 20P1 = P1. Step 9: Bring P1 terms together: -3800 = P1 - 20P1 = -19P1, so P1 = 3800 / 19 = 200. Step 10: Therefore, the initial price P1 is 200 rupees.

Verification / Alternative check:
To verify, plug P1 = 200 back into the elasticity formula. Now ΔP = 190 - 200 = -10. Then elasticity e = (ΔQ / ΔP) * (P1 / Q1) = (200 / -10) * (200 / 5000) = (-20) * (0.04) = -0.8. This matches the given elasticity exactly, confirming that the calculated initial price is correct.

Why Other Options Are Wrong:
Rs 220, Rs 240 and Rs 250 all give different values when substituted back into the elasticity formula and do not produce -0.8. They either give a different magnitude or a different sign, so they cannot be correct initial prices given the data in the question.

Common Pitfalls:
A common mistake is to ignore the negative sign of elasticity or to mix up the direction of the price change. Another frequent error is using the midpoint (arc) elasticity formula when the question clearly expects a point elasticity around the initial price. Algebra errors while solving for P1 can also lead to wrong options even if the method is correct. To avoid these pitfalls, carefully write each step, maintain sign discipline, and always verify the final answer by substituting back into the original formula.

Final Answer:
The initial price P1 that satisfies the given elasticity condition is Rs 200.