Restoring the Original Price after a Percent Reduction - A Mathematical Analysis

A scenario often encountered in market dynamics involves the adjustment of prices to restore their original values after a reduction has been applied. This article explores the mathematical techniques and calculations used to determine the necessary percentage increase required to restore a price to its original value following a discount. We will examine specific examples to understand this concept better.

Mathematical Context and Problem Framework

Let us consider the common scenario where the price of an article is reduced by a percentage. For example, if the original price of the article is 100, and it is reduced by 36%, we will determine the percentage increase required to restore it back to its original value.

Step-by-Step Calculation

Example 1:

1. **Original Price**: 100

2. **Price after 36% Reduction**: 100 - 36% of 100 100 - 36 64

Let's denote:

x as the percentage increase required to restore the price to its original value. So, the new price after the increase will be: 64(1 x/100) equal to 100 (the original price).

3. Setting up the equation:

[64(1 frac{x}{100}) 100]

4. Solving for (x):

[1 frac{x}{100} frac{100}{64}]

[frac{x}{100} frac{100}{64} - 1]

[frac{x}{100} frac{36}{64}]

[x frac{36}{64} times 100]

[x 56.25%]

Thus, the new price should be increased by 56.25% to restore it to the original value.

General Derivation

Let the original price be (P).

1. The price after a reduction of (36%) would be:

[P - 0.36P 0.64P]

Let the percentage increase needed to restore the price be (x).

2. The price after the increase would be:

[0.64P (1 frac{x}{100})]

We need this to equal (P):

[0.64P (1 frac{x}{100}) P]

3. Dividing both sides by (P) assuming (P eq 0):

[0.64 (1 frac{x}{100}) 1]

4. Solving for (x):

[1 frac{x}{100} frac{1}{0.64}]

[frac{x}{100} frac{1}{0.64} - 1]

[frac{x}{100} 1.5625 - 1]

[frac{x}{100} 0.5625]

[x 0.5625 times 100]

[x 56.25%]

Hence, the new price must be increased by 56.25% to restore it to its original value.

Additional Example

Example 2:

1. **Original Price**: 100

2. **Price after 30% Reduction**: 100 - 30% of 100 100 - 30 70

Solution:

Let's denote the percentage increase needed to restore the price to its original value as (x).

3. Equation:

[70(1 frac{x}{100}) 100]

4. Solving for (x):

[1 frac{x}{100} frac{100}{70}]

[frac{x}{100} frac{100}{70} - 1]

[frac{x}{100} frac{30}{70}]

[x frac{30}{70} times 100]

[x 42.86%]

Thus, the reduced price should be increased by 42.86% to restore it to the original value.

Summary

The above examples demonstrate that to restore the price of any article to its original value after a certain percentage reduction, one needs to calculate the required percentage increase. Utilizing algebraic manipulation, we find that for a reduction of (36%), the price needs to be increased by (56.25%), and for a reduction of (30%), the price needs to be increased by (42.86%) to restore it back to its original value.

For SEO purposes, you could use these examples to create content that explains price reductions and restoration in your product or business listings. This can help improve your page's relevance to users searching for information about price reductions and find your site when they are looking for information on restoring original prices.