Question

You are seeking to be emancipated from your parents. You are

looking for an apartment. There are two final choices.
Apartment A has a $1000 security deposit and costs $1200 each
month. Apartment B has a $1500 and costs $1175 each month.
How
many months, m, will it take for the costs to be the same? Step by step

Answer 1

20 months

Explanation:

Define the variables:

• Let m = number of months
• Let c = total cost (in dollars)

Create an equation for the total costs of renting each apartment:

• 
  `\textsf{Apartment A}:\quad c = 1000 + 1200m`

• 
  `\textsf{Apartment B}:\quad c = 1500 + 1175m`

To calculate how many months it will take for the costs to be the same, substitute the equation for Apartment A into the equation for Apartment B and solve for m:

`\implies 1000 + 1200m = 1500 + 1175m`

`\implies 1000 + 1200m - 1175m = 1500 + 1175m - 1175m`

`\implies 1000 + 25m = 1500`

`\implies 1000 + 25m - 1000 = 1500 - 1000`

`\implies 25m = 500`

`\implies 25m / 25 = 500 / 25`

`\implies m = 20`

Therefore, it will take 20 months for the costs to be the same.

Check by substituting m = 20 into both equations:

`\begin{aligned}\textsf{Apartment A}: \quad c &= 1000 + 1200(20)\\ &= 1000 + 24000\\ &= 25000\end{aligned}`

`\begin{aligned}\textsf{Apartment B}: \quad c &= 1500 + 1175(20)\\ &= 1500 + 23500\\ &= 25000\end{aligned}`

As the total cost for both apartments is $25,000 after 20 months, this proves that it will take 20 months for the costs to be the same.