Reverse Percentages

Reverse percentages allow us to determine the original value of a quantity after a percentage increase or decrease has taken place. This concept can be applied to situations such as calculating original prices before discounts or determining an initial population before an increase or decrease.

The reverse percentage formula can be applied for both increases and decreases. For a percentage increase, the formula is:

Original Value = Final Value ÷ (1 + Percentage Increase as a Decimal)

For a percentage decrease, the formula is:

Original Value = Final Value ÷ (1 – Percentage Decrease as a Decimal)

Solving Reverse Percentage Problems

When solving reverse percentage problems, follow these steps:

Step 1: Understand the problem and identify the original and final values. Carefully read the problem and determine which value is the original and which is the final value after the percentage change.

Step 2: Convert the percentage increase or decrease into a decimal. Divide the percentage by 100 to express the change as a decimal. For example, a 20% increase would be represented as 0.20.

Step 3: Calculate the original value using the reverse percentage formula. Apply the reverse percentage formula, taking into account whether the problem involves an increase or a decrease.

Let’s look at an example:

After a discount of 25%, a pair of shoes is now priced at £45. What was the original price of the shoes before the discount?

1. In this problem, we want to find the original price of the shoes (the original value) before the discount. We are given that the final price of the shoes after the discount is £45 (the final value).

2. We are given that the price decreased by 25% due to the discount. To convert the percentage into a decimal, we divide the percentage by 100:

25\% ÷ 100 = 0.25

3. For a decrease, the reverse percentage formula is:

Original value = Final value / (1 – Percentage decrease as a decimal)

Plugging in the given values, we get:

Original value = \frac{\pounds 45}{(1 - 0.25)}

= \frac{\pounds 45}{0.75}

= \pounds 60

So, the original price of the shoes before the discount was £60.

Examples

Example 1:

A jacket’s price increased by 15% and now costs £69. Calculate the original price.

Percentage increase as a decimal: 15\% \div 100 = 0.1515\% \div 100 = 0.15

Original Value = \frac{\pounds 69}{1 + 0.15} = \frac{\pounds 69}{1.15} = \pounds 60= \frac{\pounds 69}{1 + 0.15} = \frac{\pounds 69}{1.15} = \pounds 60

The original price of the jacket was £60.

Example 2:

A book’s price decreased by 10% and now costs £18. Calculate the original price.

Percentage decrease as a decimal: 10\% ÷ 100 = 0.1010\% ÷ 100 = 0.10

Original Value = \frac{\pounds 18}{1 - 0.10} = \frac{\pounds 18}{0.9} = \pounds 20= \frac{\pounds 18}{1 - 0.10} = \frac{\pounds 18}{0.9} = \pounds 20

The original price of the book was £20.

Example 3:

A store increased the price of a product by 15% during a sale, and then decreased the new price by 10% after the sale ended. The final price of the product is £102.60. What was the original price of the product?

Let the original price be £x.

Step 1: Apply the first percentage change (15% increase). New price after 15% increase: x \times (1 + 0.15) = 1.15xx \times (1 + 0.15) = 1.15x

Step 2: Apply the second percentage change (10% decrease) to the new price. Final price after 10% decrease: (1.15x) \times (1 - 0.1) = 0.9 \times 1.15x = 1.035x(1.15x) \times (1 - 0.1) = 0.9 \times 1.15x = 1.035x

Step 3: Set up an equation and solve for the original price (x). 1.035x = £102.601.035x = £102.60

Divide both sides by 1.035: x = \frac{£102.60}{1.035} = £99.13x = \frac{£102.60}{1.035} = £99.13

The original price of the product was £99.13

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