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Let's break down the problem step by step to solve it.

We are given two phone plans:

- **Plan 1**: No monthly fee but charges $0.17 per minute.
- **Plan 2**: Has a $20 monthly fee and charges $0.12 per minute.

We need to find the number of minutes (let's call it $x$) at which both plans will cost the same.

### Cost of Plan 1:
The total cost for Plan 1 is simply the cost per minute multiplied by the number of minutes:
\[
\text{Cost of Plan 1} = 0.17x
\]

### Cost of Plan 2:
The total cost for Plan 2 includes a fixed monthly fee of $20 and $0.12 per minute:
\[
\text{Cost of Plan 2} = 20 + 0.12x
\]

### Set the two costs equal:
To find when the costs are equal, we set the equations equal to each other:
\[
0.17x = 20 + 0.12x
\]

### Solve for $x$:
1. Subtract $0.12x$ from both sides:
   \[
   0.17x - 0.12x = 20
   \]
   \[
   0.05x = 20
   \]

2. Divide both sides by 0.05:
   \[
   x = \frac{20}{0.05} = 400
   \]

So, the two plans will cost the same when the customer uses **400 minutes** of calls.

Would you like more details or have any questions?

Here are five related questions you could explore:
1. What would the cost of each plan be for 500 minutes of calls?
2. If the monthly fee for Plan 2 increases to $25, how would that change the number of minutes at which the plans are equal?
3. How would the answer change if Plan 1 charged $0.20 per minute instead of $0.17?
4. What is the total cost for Plan 2 if no minutes are used?
5. How would the total cost for each plan change if only 300 minutes were used?

**Tip:** When comparing two different plans with a fixed cost and a variable cost, it's often helpful to set up an equation and solve for the point where the costs are equal to make an informed decision!

Customers of a phone company can choose between two service plans for long distance calls. The first plan has no monthly fee but charges $0.17 for each minute of calls. The second plan has a $20 monthly fee and charges an additional $0.12 for each minute of calls. For how many minutes of calls will the costs of the two plans be equal?

Learn how to solve a real-world problem comparing phone plans. We find the number of minutes of calls at which two different plans (one with a fixed monthly fee and another with a per-minute charge) cost the same. The solution uses linear equations and algebra, making it suitable for Grades 7-10.

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