one taxi company charges 75 cents for the first quarter mile and 15 cents for each additional mile the competing taxi company charges 1.00 for the first quarter mile and 10 each additional mile what distance would produce the same fare for the two taxi companies?

The fare is the same when the distance x satisfies this equation:

0.75 + 0.15(x-0.25) = 1.00 + 0.10(x-0.25)

0.15x + 0.7125 = 0.10x + 0.9750

x = 5.25 miles

You could do this without algebra. After the first quarter mile, the second taxi costs 25 cents more. You need to go 5 more miles to make up the difference at 5 cents per mile.

To determine the distance that would produce the same fare for the two taxi companies, we need to set up an equation and solve for the distance.

Let's denote the distance in miles as 'd'.

For the first taxi company, the fare is calculated as follows:
- 75 cents for the first quarter mile (0.25 miles)
- 15 cents for each additional mile (d - 0.25)

So the fare for the first taxi company is:
F1 = 0.75 + 0.15 * (d - 0.25)

For the second taxi company, the fare is calculated as follows:
- 1.00 for the first quarter mile (0.25 miles)
- 10 cents for each additional mile (d - 0.25)

So the fare for the second taxi company is:
F2 = 1.00 + 0.10 * (d - 0.25)

To find the distance that would produce the same fare for both companies, we can set up the equation: F1 = F2

0.75 + 0.15 * (d - 0.25) = 1.00 + 0.10 * (d - 0.25)

Now, we can solve this equation for 'd':
0.75 + 0.15d - 0.0375 = 1.00 + 0.10d - 0.025

Combining like terms:
0.15d + 0.7125 = 1.00 + 0.10d - 0.025

Subtracting 0.10d from both sides:
0.05d + 0.7125 = 1.00 - 0.025

Subtracting 0.7125 from both sides:
0.05d = 0.2875

Dividing both sides by 0.05:
d = 5.75

Therefore, the distance that would produce the same fare for both taxi companies is 5.75 miles.