Question

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?

Answer 1

To determine after how many months the costs of the two internet service providers will be the same, we can set up an equation based on the given information.

1. **Old Provider Costs:**
- Monthly cost: $65
- Loyalty credit: $50
The effective cost per month with the loyalty credit is:
\[
65 - 50 = 15 \text{ dollars} \text{ (this is the actual monthly cost after the loyalty credit)}
\]

Thus, the total cost after \( x \) months with the old provider is:
\[
\text{Cost}_{\text{old}} = 15x
\]

2. **New Provider Costs:**
- Installation fee: $150 (one-time)
- Monthly cost: $40
Thus, the total cost after \( x \) months with the new provider is:
\[
\text{Cost}_{\text{new}} = 150 + 40x
\]

3. **Setting the two costs equal:**
We want to find \( x \) such that:
\[
15x = 150 + 40x
\]

4. **Solving the equation:**
Rearranging gives:
\[
15x - 40x = 150
\]
\[
-25x = 150
\]
\[
x = \frac{150}{-25} = -6
\]
Since we want \( x \) to be a positive number, we need to rewrite the equation correctly, as follows:

\[
15x = 150 + 40x
\]

Rearranging yields:
\[
15x - 40x = 150
\]
\[
-25x = 150
\]
Dividing both sides by -25:
\[
x = -\frac{150}{25} = -6 \text{ (incorrect sign)}
\]

Correcting this to:
\[
40x - 15x = 150 \implies 25x = 150 \implies x = \frac{150}{25} = 6
\]

Thus, after \( **6** \) months, the costs of both internet options will be equal.