How do you calculate back from a percentage?

Unraveling the Percentage Puzzle: Finding the Original Value

Percentages are everywhere. Discounts in stores, interest rates on loans, even the charge level on your phone – they paint a picture of proportions and changes. But what if you know the altered value after a percentage has been applied and need to find the original amount? Understanding how to “calculate back” from a percentage is a valuable skill, unlocking the secrets hidden behind these ubiquitous numbers.

While the concept might seem intimidating at first, the method is surprisingly straightforward. Let’s break down the process into easily digestible steps:

1. Determine the Remaining Percentage:

This is the crucial first step. The key is to understand whether the percentage represents an increase or a decrease to the original value.

• For an Increase (e.g., a markup, sales tax): Add the given percentage to 100%. This tells you what percentage of the original amount the new value represents. For example, if an item has a 20% markup, the new price is 120% of the original price.
• For a Decrease (e.g., a discount, sale): Subtract the given percentage from 100%. This tells you what percentage of the original amount is left after the reduction. For example, if an item is 30% off, the sale price is 70% of the original price.

2. Find What Represents 1%:

Now that you know the relationship between the altered value and the percentage of the original it represents, you can find the value of 1%. To do this, simply divide the altered value (the value you know after the percentage change) by the percentage you calculated in step one.

Formula: 1% Value = Altered Value / Remaining Percentage

For instance, if an item costs $150 after a 20% markup (meaning $150 represents 120% of the original price), you would divide $150 by 120 to find what 1% represents.

3. Calculate the Original Amount (100%):

The final step is to find the original amount, which represents 100%. Since you now know the value of 1%, simply multiply that value by 100.

*Formula: Original Amount = 1% Value 100**

Continuing our example, you would multiply the value of 1% (calculated from the previous step) by 100 to find the original price of the item.

Let’s Illustrate with Examples:

• Example 1: Finding the Original Price After a Discount

  A shirt is on sale for $35, which is 30% off the original price. What was the original price?

  1. Remaining Percentage: 100% – 30% = 70%
  2. 1% Value: $35 / 70 = $0.50
  3. Original Amount: $0.50 * 100 = $50

  Therefore, the original price of the shirt was $50.

• Example 2: Finding the Original Price After Sales Tax

  You paid $108 for a product, including 8% sales tax. What was the original price of the product?

  1. Remaining Percentage: 100% + 8% = 108%
  2. 1% Value: $108 / 108 = $1
  3. Original Amount: $1 * 100 = $100

  Therefore, the original price of the product was $100.

Key Takeaways:

• Always identify whether the percentage represents an increase or a decrease.
• Remember that 100% represents the original, unknown value.
• The formula “Altered Value / Remaining Percentage = 1%” is the cornerstone of this calculation.

Calculating back from a percentage might initially seem complex, but by understanding the underlying principles and breaking it down into manageable steps, you can confidently unravel any percentage-based problem and reveal the original values hidden within. Practice with different scenarios, and soon you’ll be a percentage pro!