Answer:
20 months
Step-by-step 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:
- [tex]\textsf{Apartment A}:\quad c = 1000 + 1200m[/tex]
- [tex]\textsf{Apartment B}:\quad c = 1500 + 1175m[/tex]
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:
[tex]\implies 1000 + 1200m = 1500 + 1175m[/tex]
[tex]\implies 1000 + 1200m - 1175m = 1500 + 1175m - 1175m[/tex]
[tex]\implies 1000 + 25m = 1500[/tex]
[tex]\implies 1000 + 25m - 1000 = 1500 - 1000[/tex]
[tex]\implies 25m = 500[/tex]
[tex]\implies 25m \div 25 = 500 \div 25[/tex]
[tex]\implies m = 20[/tex]
Therefore, it will take 20 months for the costs to be the same.
Check by substituting m = 20 into both equations:
[tex]\begin{aligned}\textsf{Apartment A}: \quad c &= 1000 + 1200(20)\\ &= 1000 + 24000\\ &= 25000\end{aligned}[/tex]
[tex]\begin{aligned}\textsf{Apartment B}: \quad c &= 1500 + 1175(20)\\ &= 1500 + 23500\\ &= 25000\end{aligned}[/tex]
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.
