# Price after Discount

How to solve price after discount problems: formulas, proofs, examples, and their solutions.

## Formula 1: Discount

(discount) = (price)⋅(discount rate)

(discount): Amount of discount
(price): Original price
(discount rate): The rate without any unit
(i.e. 10% → 0.10)

## Proof 1

The discount rate is the ratio of (discount)/(price).

Ratio

Multiply (price) on both sides.

Switch both sides.

Then you get
(discount) = (price)⋅(discount rate).

## Example 1

(original price) = 200
(discount rate) = 0.30

So (discount) = 200⋅0.30
= 60.

The unit of the discount is \$.

So write the answer with its unit: \$60.

## Formula 2: Price after Discount

(final) = (price)[1 - (discount rate)]

(final): Price after discount
(price): Original price

By using this formula,
you can directly find the price after discount.

## Proof 2

The price after discount can be found by
subtracting the amount of discount
from the original price.

So (final) = (price) - (discount).

Change (price) to (price)⋅1.

And you know that
(discount) = (price)⋅(discount rate).

Change (price)⋅1 - (price)⋅(discount rate)
to (price)[1 - (discount rate)].

This trick, which is called factoring, is also used
when proving the price after sales tax formula.

Just think this trick as
2⋅1 - 2⋅3 = 2(1 - 3).

Price after sales tax - Proof of the formula

Factoring - Using the distributive property

So the price after discount, final, is
(final) = (price)[1 - (discount)].

## Example 2

(original price) = 200
(discount rate) = 0.30

So (final) = 200(1 - 0.30)
= 140.

The unit of the price is \$.

So write the answer with its unit: \$140.