The price of rice has increased by 60 percent. By what percentage must the new price be reduced so that the original price is restored?

Introduction / Context:
This question focuses on successive percentage changes, specifically an increase followed by a decrease. It is important to note that a 60 percent increase cannot be simply reversed by a 60 percent decrease, because the bases of these percentages are different. The goal is to find the exact percentage reduction needed to restore the original price of rice.

Given Data / Assumptions:
Original price of rice = P (some positive amount). Price is first increased by 60 percent. Let the new price be N. We must find the percentage decrease from N that brings the price back to P.

Concept / Approach:
A 60 percent increase means the new price is P * (1 + 60/100) = 1.6P. Suppose the required reduction percentage is x percent. Then the reduced price is N * (1 - x/100). We need N * (1 - x/100) to equal P. Substituting N = 1.6P gives an equation in x, which we can solve to find the required percentage decrease.

Step-by-Step Solution:
First increase: N = P * (1 + 60/100) = P * 1.6. Let required reduction percentage be x percent. After reduction, price becomes N * (1 - x/100). This final price must equal original price P. So N * (1 - x/100) = P. Substitute N = 1.6P: 1.6P * (1 - x/100) = P. Divide both sides by P (P is non zero): 1.6 * (1 - x/100) = 1. So 1 - x/100 = 1 / 1.6. Compute 1 / 1.6 = 10 / 16 = 5 / 8 = 0.625. Thus 1 - x/100 = 0.625. So x/100 = 1 - 0.625 = 0.375. Therefore x = 0.375 * 100 = 37.5 percent. Write 37.5 percent as 37 1/2 percent.

Verification / Alternative check:
Assume original price P = 100 units. After a 60 percent increase, new price N = 160. A reduction of 37.5 percent on 160 equals 37.5/100 * 160 = 60. The price after reduction becomes 160 - 60 = 100, which matches the original price. This numerical example confirms that 37.5 percent is the correct required reduction.

Why Other Options Are Wrong:
A reduction of 33 1/3 percent on 160 would lower the price by about 53.33, giving 106.67, not 100. A 40 percent reduction would lower 160 by 64, giving 96. A reduction of 45 percent would drop the price to 88. Fifty percent reduction would cut it to 80. None of these restore the original price of 100. Only 37 1/2 percent works exactly.

Common Pitfalls:
Many learners assume that a 60 percent increase followed by a 60 percent decrease brings the price back to the start, which is incorrect because the decrease is based on the larger price. Another mistake is to subtract percentages directly without considering their different bases. Setting up the equation carefully avoids such misunderstandings.

Final Answer:
The new price must be reduced by 37 1/2 percent to restore the original price of rice.