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: \[ 1.06p \text{ dollars} \]