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I have a database of prices that already have a 10% discount.

For example a product could be $100 after a 10% discount. Is there a reusable formula I can use to determine what the original price was of all the 10% discounted prices in the database?

Edit: Thank you for the fast responses. Is there any way to account for rounding errors? A real example is a product with a discounted price of \$129.00 Using the X/.9 formula, I get \$143.33 as the original price, which does not actually work out. To have had \$129.00 as the discount price, the original price would have needed to have been \$143.34.

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Divide by $0.9$. To check that this works, observe that e.g. $\frac{180}{0.9}=200$, so the original price belonging to a $180\$$ reduced price is $200\$$. (the divisor $0.9$ is just a $10\%$ “reduced” $1$).

More general:

- Let $d$ denote the
*discount*in % - let $p_{orig}$ be the original price
- and let $p_{disc}$ denote the discount price

Then it holds

$$\left(\frac{100-d}{100}\right)\cdot p_{orig} = p_{disc}$$

With this formula you are able to calculate the prices even if there is another discount offered.

In your special case you have $d=10$ thus the formula leeds to:

$$\left(\frac{100-10}{100}\right)\cdot p_{orig} = p_{disc}$$

$$\Longrightarrow 0.9\cdot p_{orig} = p_{disc}$$

Let say the price of something is $x$. Giving a discount of 10% means that you decrease the price to 90% of the original. That means that after taking 10% off, you have 90% of $x$ left. That is $0.90x$.

In your concrete problem you know that $0.90x = 100$.

And you just need to solve this equation to find the original price $x$.

Original price $=\frac{10}{9}\times$ Discounted price.

Remember that a 10% discount means that it costs 90% of the original price. So you multiply the original price by 0.90.

Let $O$ be the original price, and $N$ the new price.

So $0.9\cdot O = N$

So $O = \frac{N}{0.9}$

As pointed out in another answer

More general:

- Let $d$ denote the
*discount*in % - let $p_{orig}$ be the original price
- and let $p_{disc}$ denote the discount price

Then it holds

$$\frac{100-d}{100}\cdot p_{orig} = p_{disc}$$

So we can rearrange this to get

$$p_{orig} = p_{disc} \cdot \frac{100}{100-d}$$

Which lets you find the original price as required.

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