Question

A theater group made appearances in two cities. The hotel charge before tax in the second city was $500 higher than in the first. The tax in the first city was5%, and the tax in the second city was 6%. The total hotel tax paid for the two cities was $552.50. How much was the hotel charge in each city before tax?Note that the ALEKS graphing calculator can be used to make computations easier.

Answer 1

Let us represent the hotel charge with different variables x and y:

Let the hotel charge before tax in the first city be x

Let the hotel charge before tax in the second city be y

Now, let us represent the word problem in equation form:

First, we were told that the charge before tax in the second city is $500 more than the charge before tax in the first city, this can be represented thus:

`y=x+500\ldots\text{.eqn 1}`

Going forward in the question, we were told the tax for the first city (x) is 5%(0.05), and the tax for the second city is 6%(0.06). The total tax from both cities is $552.5, this expression can be written mathematically as:

`0.05x+0.06y=552.5\ldots\ldots\text{eqn 2}`

Now, by solving equation 1 and equation 2 simultaneously, we will obtain the hotel charge in each city before tax. (that is the value of x and y).

`\begin{gathered} y=x+500 \\ 0.05x+0.06y=552.5 \\ \end{gathered}`

Using, the substitution method of solving simultaneous equation, we will solve further:

`\begin{gathered} \text{substitute equation 1 into equation 2} \\ 0.05x+0.06(x+500)=552.5 \\ 0.05x+0.06x+30=552.5 \\ 0.11x+30=552.5 \end{gathered}`
`\begin{gathered} 0.11x=552.5-30 \\ 0.11x=522.5 \\ x=(522.5)/(0.11) \\ x=4750 \end{gathered}`

The hotel charge before tax in the first city is $4750.

Now, substitute the value of x into equation 1 to get the value of y (hotel charge before tax in the second city)

`\begin{gathered} y=x+500 \\ x=4750 \\ y=4750+500 \\ y=5250 \end{gathered}`

The hotel charge before tax in the second city is $5250.