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 was 3.5%, and the tax in the second city was 5.5%. The total hotel tax paid for the two cities was $365. How much was the hotel charge in each city before tax? First city: Second city:

Answer 1

Let's call the hotel charge in the first city as x. The tax in the first city was 3.5%, therefore, if we call the tax paid in the first city as t1, the tax paid is given by the following expression:

`t_1=0.035x`

The hotel charge before tax in the second city was $500 higher than in the first and the tax in the second city was 5.5%, therefore, if we call the tax paid in the second city as t2, the tax paid is given by the following expression:

`t_2=0.055(x+500)=0.055x+27.5`

If we multiply the first equation by 0.055/0.035, we're going to have

`\begin{gathered} t_1\cdot(0.055)/(0.035)=0.035x\cdot(0.055)/(0.035) \\ (11)/(7)t_1=0.055x \end{gathered}`

Then, if we subtract this expression from the expression for the tax on the second city, we're going to have

`\begin{gathered} t_2-((11)/(7)t_1)=0.055x+27.5-(0.055x) \\ t_2-(11)/(7)t_1=27.5 \end{gathered}`

The total hotel tax paid for the two cities was $365. This gives to us another equation

`t_1+t_2=365`

If we subtract this equation from the previous expression, we're going to have a new equation only for the tax of the first city.

`\begin{gathered} t_2-(11)/(7)t_1-(t_1+t_2)=27.5-(365) \\ t_2-(11)/(7)t_1-t_2-t_1=27.5-365 \\ -(18)/(7)t_1=-337.5 \\ t_1=(7)/(18)\cdot337.5 \\ t_1=131.25 \end{gathered}`

Using this value on the first equation, we're able to determinate the value of x.

`\begin{gathered} t_1=0.035x \\ (131.25)=0.035x \\ x=(131.25)/(0.035) \\ x=3750 \end{gathered}`

The hotel charge on the first city before tax was $3750.00.

Since the hotel charge before tax in the second city was $500 higher than in the first, we have

`x+500=3750+500=4250`

The hotel charge on the second city before tax was $4250.00.