Price after Sales Tax

Price after Sales Tax

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

Formula 1: Sales Tax

(sales tax) = (price)*(sales tax rate)

The sales tax, or the value-added tax (VAT),
is a tax that is added to the price of a product.

(sales tax) = (price)⋅(sales tax rate)

(sales tax): Amount of sales tax
(price): Price before sales tax
(discount rate): The rate without any unit
(i.e. 10% → 0.10)

Proof 1

Sales Tax: Proof of the Formula

The sales tax rate is the ratio of (sales tax)/(price).

Ratio

Multiply (price) on both sides.

Switch both sides.

Then you get
(sales tax) = (price)⋅(sales tax rate).

Example 1

The price of a product before sales tax is $400. If the sales tax rate is 8%, find the amount of sales tax.

(original price) = $400
(sales tax rate) = 0.08 (not 8!)

So (sales tax) = 400⋅0.08
= 32.

The unit of the sales tax is $.

So write the answer with its unit: $32.

Formula 2: Price after Sales Tax

(final) = (price)[1 + (sales tax rate)]

(final) = (price)[1 + (sales tax rate)]

(final): Price after sales tax
(price): Price before sales tax

By using this formula,
you can directly find the price after sales tax.
(= the value-added tax, VAT)

Proof 2

Price after Sales Tax: Proof of the Formula

The price after sales tax
is the sum of the price before sales tax
and the amount of its sales tax.

So (final) = (price) + (sales tax).

Change (price) to (price)⋅1.

And you know that
(sales tax) = (price)⋅(sales tax rate).

Change (price)⋅1 + (price)⋅(sales tax rate)
to (price)[1 + (sales tax rate)].

This trick is factoring,
which you'll learn later.

So just think this trick as
2⋅1 + 2⋅3 = 2(1 + 3).

Factoring - Using the distributive property

So the price after sales tax, final, is
(final) = (price)[1 + (sales tax rate)].

Example 2

The price of a product before sales tax is $400. If the sales tax rate is 8%, find the price after sales tax.

(original price) = $400
(sales tax rate) = 0.08 (again, not 8!)

So (final) = 400(1 + 0.08)
= 432.

The unit of the price is $.

So write the answer with its unit: $432.