A phone company offers two monthly plans. Plan A costs

US$22

plus an additional

US$0,11

for each minute of calls. Plan B costs

US$14

plus an additional

US$0,15

for each minute of calls.

For what amount of calling do the two plans cost the same?

What is the cost when the two plans cost the same?

Let's define the number of minutes of calls as $ x $.

### Plan A:
- Fixed cost: \$22
- Variable cost: \$0.11 per minute

The total cost for Plan A can be expressed as:
\[
\text{Cost of Plan A} = 22 + 0.11x
\]

### Plan B:
- Fixed cost: \$14
- Variable cost: \$0.15 per minute

The total cost for Plan B can be expressed as:
\[
\text{Cost of Plan B} = 14 + 0.15x
\]

### Finding the amount of calling where the two plans cost the same:
To find the number of minutes where both plans cost the same, we set the two cost expressions equal to each other and solve for $ x $:

\[
22 + 0.11x = 14 + 0.15x
\]

Subtract 14 from both sides:
\[
8 + 0.11x = 0.15x
\]

Subtract $ 0.11x $ from both sides:
\[
8 = 0.04x
\]

Solve for $ x $ by dividing both sides by 0.04:
\[
x = \frac{8}{0.04} = 200
\]

So, the two plans cost the same when you make 200 minutes of calls.

### Cost when the two plans cost the same:
To find the cost, substitute $ x = 200 $ into either cost equation (both should give the same result):

Using Plan A's cost equation:
\[
\text{Cost of Plan A} = 22 + 0.11(200) = 22 + 22 = 44
\]

So, when the two plans cost the same, the cost is **\$44**.

---

Would you like more details on this solution or have any other questions? Here are some related questions to explore:

1. How would the costs change if Plan B had a lower per-minute rate?
2. What if Plan A had a lower fixed cost?
3. How many minutes should one talk to make Plan A cheaper than Plan B?
4. How do the costs differ if the number of minutes is 100?
5. How would you graph the cost of both plans?
6. What if a third plan was introduced with a different cost structure?
7. How does a change in fixed cost affect the break-even point?
8. How would rounding of costs to the nearest cent affect the results?

**Tip:** When comparing costs in similar scenarios, it's often useful to express both cost structures algebraically and find the intersection point for a clear comparison.

Learn how to find the break-even point between two phone plans A and B with a step-by-step solution. Understand the cost analysis involving linear equations and fixed plus variable costs. Suitable for Grades 7-9 students.

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