To find the fixed fee the mechanic charges, you can set up an equation based on the information provided. Let F represent the fixed fee, and R be the hourly rate. We know that for 4 hours of service, the cost is $270, so we can write the equation as:

4R + F = 270

And for 7 hours of work, the cost is $420, which gives us another equation:

7R + F = 420

Now, you can solve these two equations to find the values of F (the fixed fee) and R (the hourly rate). Once you find the fixed fee (F), you'll have the answer to the question