United Telephone charges a base rate of `$10.00` for service, plus an additional charge of `$0.25` per minute. Atlantic Call 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 solve the problem, we need to set up equations for the charges from both telephone companies and find the point at which they are equal.

### Step-by-Step Solution:

1. **Define Variables**:
Let \( x \) be the number of minutes for which the bills of both companies are the same.

2. **Set Up the Equation for United Telephone**:
United Telephone charges a base rate of $10.00 plus $0.25 per minute. Therefore, the total charge \( C_U \) for United Telephone can be expressed as:
\[ C_U = 10 + 0.25x \]

3. **Set Up the Equation for Atlantic Call**:
Atlantic Call charges a base rate of $12.00 plus $0.20 per minute. Therefore, the total charge \( C_A \) for Atlantic Call can be expressed as:
\[ C_A = 12 + 0.20x \]

4. **Set the Equations Equal to Each Other**:
To find the number of minutes \( x \) where the charges are the same, we set \( C_U \) equal to \( C_A \):
\[ 10 + 0.25x = 12 + 0.20x \]

5. **Rearrange the Equation**:
To solve for \( x \), first, we can move all terms involving \( x \) to one side and constant terms to the other side:
\[ 0.25x - 0.20x = 12 - 10 \]
Simplifying this gives:
\[ 0.05x = 2 \]

6. **Solve for \( x \)**:
Now, divide both sides by \( 0.05 \) to isolate \( x \):
\[ x = \frac{2}{0.05} \]
Calculating this gives:
\[ x = 40 \]

7. **Conclusion**:
The number of minutes for which the bills for each telephone company would be the same is \( 40 \) minutes.