Mathematical Problem Solving: Calculating the Original Price of a Bicycle After Multiple Percentage Increases

Problems involving percentage increases and finding original values or costs are quite common, especially in the realm of sales, pricing, and finance. This article focuses on a specific scenario involving multiple percentage increases on the price of a bicycle, followed by the calculation of its original price. This step-by-step guide will help you solve such problems.

Understanding the Problem

The problem presented involves a bicycle whose price undergoes several percentage increases. Specifically, the price increases by 9.09%, 8.33%, and 7.7%. The final price after these increases is Rs. 1274. The goal is to determine the original price of the bicycle before these increases.

Step-by-Step Solution

To solve this problem, let's denote the original price of the bicycle as x. The price increases sequentially, and each increase can be represented as a multiplication by a factor.

First Increase of 9.09%

The first increase of 9.09% means that the price becomes 109.09% of its original value. In fractional form, this is expressed as 1.0909. Therefore, after the first increase, the price is:

x × 1.0909 1.0909x

Second Increase of 8.33%

The second increase of 8.33% means that the price becomes 108.33% of its value after the first increase. In fractional form, this is expressed as 1.0833. Therefore, after the second increase, the price is:

(1.0909x) × 1.0833 1.0909 × 1.0833x 1.1817x

Third Increase of 7.7%

The third increase of 7.7% means that the price becomes 107.7% of its value after the second increase. In fractional form, this is expressed as 1.077. Therefore, after the third increase, the price is:

(1.1817x) × 1.077 1.1817 × 1.077x 1.274x

Setting Up the Equation

We are given that the final price after all these increases is Rs. 1274. Therefore, we can set up the equation as follows:

1.274x 1274

Solving for x, we get:

x 1274 / 1.274 1000

Conclusion

The original price of the bicycle was Rs. 1000.

Further Considerations

This problem can also be analyzed from another perspective. If the last increase were to be 7.14%, which is an increase of 1/14, the calculation would be slightly different. Let's consider that final increase:

Calculation with 7.14% Increase

If the last increase were 7.14%, the price after this increase would be 107.14% or 1 7.14/100 107.14/100 1.0714. Therefore, the equation would be:

x × (12/11) × (13/12) × (107.7/100) 1274

If the last increase were 7.14%, the equation would be:

x × (12/11) × (13/12) × (15/14) 1274

These calculations can be performed using a calculator or algebraic methods to find the original value x.

In conclusion, understanding and solving problems involving multiple percentage increases is a valuable skill in many fields, including business, finance, and economics. The step-by-step approach demonstrated here can be applied to similar types of problems to find the original values or costs.