The price of a fan is decreased by 40% and its sales increase by 50%. What is the overall percentage change in revenue?

Introduction / Context:
This question relates price changes and sales volume changes to the total revenue of a seller. When price per unit decreases but the quantity sold increases, the final effect on revenue depends on the product of the new price and the new quantity. This kind of analysis frequently appears in economics and business mathematics, particularly when evaluating the impact of discounts or promotions.

Given Data / Assumptions:
Initial price of the fan is P (unknown but not needed explicitly).
Initial quantity sold is Q units.
Price decreases by 40 percent, so new price is lower.
Sales volume increases by 50 percent, so new quantity is higher.
We must find the net percentage change in revenue (price * quantity).

Concept / Approach:
Revenue is given by R = price * quantity. Let initial revenue be R0 = P * Q. After a 40 percent price decrease, the new price is P * (1 - 0.40) = 0.60 * P. After a 50 percent increase in quantity, the new quantity is Q * (1 + 0.50) = 1.50 * Q. So new revenue R1 = 0.60 * P * 1.50 * Q = 0.90 * P * Q. This means the new revenue is 90 percent of the original revenue, resulting in a 10 percent decrease overall. We can express the percentage change as (R1 - R0) / R0 * 100.

Step-by-Step Solution:
Let initial price be P and initial quantity be Q. Initial revenue R0 = P * Q. Price decreases by 40 percent, so new price = P * (1 - 0.40) = 0.60 * P. Sales increase by 50 percent, so new quantity = Q * (1 + 0.50) = 1.50 * Q. New revenue R1 = (0.60 * P) * (1.50 * Q). Compute the factor: 0.60 * 1.50 = 0.90. So R1 = 0.90 * P * Q = 0.90 * R0. This means the new revenue is 90 percent of the original revenue. Percentage change in revenue = (R1 - R0) / R0 * 100 = (0.90R0 - R0) / R0 * 100. Compute (0.90R0 - R0) / R0 = -0.10, so percentage change = -10 percent. Thus, revenue decreases by 10 percent.

Verification / Alternative check:
We can verify with simple numbers. Suppose the original price P = 100 and quantity Q = 10. Then R0 = 100 * 10 = 1000. New price after 40 percent reduction is 60. New quantity after 50 percent increase is 15. New revenue R1 = 60 * 15 = 900. The decrease in revenue is 1000 - 900 = 100, which is 10 percent of 1000. This confirms the 10 percent decrease result.

Why Other Options Are Wrong:
A 15 percent increase or decrease would imply new revenue at 115 percent or 85 percent of the original, which does not match the factor 0.90 we derived. A 10 percent increase would imply revenue of 110 percent of the original, contradicting the fact that we clearly have 0.90 times the previous revenue. Only a 10 percent decrease correctly represents the net effect of the 40 percent price cut and 50 percent rise in quantity sold.

Common Pitfalls:
Some students mistakenly subtract 40 percent and add 50 percent to get a net 10 percent increase, which is incorrect because price and quantity affect revenue multiplicatively, not additively. Others may forget to convert percentage changes into decimal multipliers before multiplying. To avoid mistakes, always treat each percentage change as a multiplicative factor, multiply the factors, and then compare the product with 1 to determine the net percentage change.

Final Answer:
The overall revenue experiences a 10% decrease.