Finding Original Prices Using Equations

Property To find the original price when you know the final price after a discount or markup, set up an equation. Let a variable represent the unknown original price and solve for it. For a discount: $$\text{Original Price} \times (1 \text{discount decimal}) = \text{Sale Price}$$. For a markup: $$\text{Original Price} \times (1 + \text{markup decimal}) = \text{Final Price}$$.

Examples A shirt is on sale for 36 dollars after a 25% discount. To find the original price $p$, solve the equation $p(1 0.25) = 36$, or $0.75p = 36$. Dividing by 0.75 gives $p = 48$ dollars. A store sells a jacket for 180 dollars after marking it up 20% from wholesale. The wholesale price $w$ can be found with $w(1 + 0.20) = 180$, or $1.20w = 180$. The wholesale price was $w = 150$ dollars. A video game costs 42 dollars after a 30% discount. To find the original price $x$, solve $x(1 0.30) = 42$, or $0.70x = 42$. The original price was $x = 60$ dollars.

Explanation Sometimes you know the sale price after a discount or the final price after a markup, but need to find the original price. You can work backward using an equation. If a 25% discount resulted in a 75 dollars sale price, you know that 75% of the original price equals 75 dollars.