Question

Arthur wants to buy an item that costs p dollars before tax. Using a​ 6% sales tax​ rate, write two different expressions that represent the price of the item after tax. Show that the two expressions are equal.

Answer 1

To determine the price of the item after including a 6% sales tax on a cost of \( p \) dollars, we can formulate two different expressions.

### Expression 1: Based on the tax rate
The sales tax can be calculated as 6% of the price \( p \). Therefore, the tax amount is:
\[
0.06p
\]
The total price after including the tax is then:
\[
p + 0.06p
\]
We can factor this expression:
\[
p + 0.06p = (1 + 0.06)p = 1.06p
\]

### Expression 2: Based on the total after applying the increment
We can also express the price after tax directly by multiplying the original price \( p \) by the total percentage (which is 106% or 1.06 when expressed as a decimal):
\[
p \times 1.06
\]

Both expressions represent the same total price after applying the 6% sales tax.

### Showing that the two expressions are equal:
From Expression 1, we found:
\[
1.06p
\]
From Expression 2, we also have:
\[
1.06p
\]

Since both expressions equal \( 1.06p \), we can conclude that:
\[
p + 0.06p = p \times 1.06
\]

Thus, the two expressions are equal, and Arthur can use either expression to determine the price of the item after tax. The final price after tax is:
\[
1.06p \text{ dollars}
\]