United Telephone charges a base rate of `$10.00` for service, plus an additional charge of `$0.25` per minute. Atlantic Cell charges a base rate of `$12.00` for service, plus an additional charge of `$0.20` per minute. For what number of minutes would the bills for each telephone company be the same?

Answer

To find the number of minutes for which the bills for United Telephone and Atlantic Cell are the same, we will set up equations based on their respective charging structures and then solve for the number of minutes.

### Step-by-step Solution:

1. **Define the Charges for Each Company:**
- United Telephone charges a base rate of $10.00 plus $0.25 per minute.
- The total charge for United Telephone after x minutes is:
\[ \text{Charge}_{\text{United}} = 10 + 0.25x \]

- Atlantic Cell charges a base rate of $12.00 plus $0.20 per minute.
- The total charge for Atlantic Cell after x minutes is:
\[ \text{Charge}_{\text{Atlantic}} = 12 + 0.20x \]

2. **Set the Charges Equal:**
- We need to find the value of x where both charges are equal:
\[ 10 + 0.25x = 12 + 0.20x \]

3. **Rearrange the Equation:**
- To solve for x, first, we will move all terms involving x to one side and constant terms to the other side:
\[ 0.25x - 0.20x = 12 - 10 \]

4. **Simplify the Equation:**
- This simplifies to:
\[ 0.05x = 2 \]

5. **Solve for x:**
- Now, divide both sides by 0.05 to find x:
\[ x = \frac{2}{0.05} \]

6. **Calculate the Value of x:**
- To simplify \(\frac{2}{0.05}\):
\[ x = \frac{2}{\frac{5}{100}} = 2 \times \frac{100}{5} = 2 \times 20 = 40 \]

7. **Conclusion:**
- Therefore, the number of minutes for which the bills for both companies are the same is:
\[ \boxed{40} \]