Explanation:
The relevant relationships are ...
discount = (discount rate) × (original price)
(sale price) = (original price) - discount
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Putting these into one equation gives ...
(sale price) = (original price) - (discount rate)×(original price)
Combining terms, yields ...
(sale price) = (original price)×(1 -(discount rate))
This relation can be solved for either of the variables on the right:
(original price) = (sale price)/(1 -(discount rate))
(discount rate) = 1 -(sale price)/(original price)
or ...
(discount rate) = discount/(original price)
Going back to the beginning, you have ...
discount = (original price) -(sale price)
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To solve for the missing amount, you can choose the formula for which you have the other amounts. Or, you can start with any of the relations that relate the amounts you have, fill in the given values, and solve for the unknown.
It's not rocket science. You see these calculations almost every time you go to the store.
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<em>Additional comments</em>
Here, we have assumed a positive value for discount rate and discount, and that the sale price is less than the original price. This is basically the set of relations that are seen by a retail <em>buyer</em>.
If you are the <em>seller</em>, the relations you typically see are ...
(cost price) + (markup) = (selling price)
The markup can be expressed as a percentage of either the cost price or the selling price, or it may be based on some other relation (fixed business costs, for example).
And another one is ...
(markup)/(cost price) = (profit ratio) . . . . often expressed as a percentage
Sometimes the profit is expressed as a fraction of the selling price (business revenue) instead of the cost price.
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Taxes are added on in the same way that discounts are subtracted:
(final selling price) = (selling price) + (tax rate)×(selling price)
or
(final selling price) = (selling price)×(1 +(tax rate))
