Bob is looking for internet service for his new apartment. If he continues with his old internet provider, he will pay $65 per month and receive a $50 customer loyalty credit. The new internet provider he is considering will charge a one-time installation fee of $150, plus $40 per month. After how many months would the two options cost the same? months

Answer 1

To find out after how many months Bob's costs for his old internet provider and the new internet provider will be the same, we can set up an equation for the total cost over time for both providers.

1. Old Provider Costs:

   • Monthly cost: $65
   • Loyalty credit: $50 (this can be treated as a discount for the first month)
   • Total cost after \( n \) months with loyalty credit for the first month: \[ \text{Total Cost Old} = 65n - 50 \quad \text{(first month discounted)} \] Note: the $50 credit applies only to the first month.

2. New Provider Costs:

   • Installation fee: $150 (one-time fee)
   • Monthly cost: $40
   • Total cost after \( n \) months: \[ \text{Total Cost New} = 150 + 40n \]

Now, we want to find \( n \) such that the total costs are equal: \[ 65n - 50 = 150 + 40n \]

To solve for \( n \):

1. Rearrange the equation: \[ 65n - 40n = 150 + 50 \] \[ 25n = 200 \]

2. Divide both sides by 25: \[ n = \frac{200}{25} = 8 \]

Thus, after 8 months, the costs of both providers will be the same.