Let \( x \) be the number of miles Arun traveled in the taxi. The total fare charged by the taxi company consists of a pick-up fee and a charge per mile:
\[ \text{Total fare} = \text{pick-up fee} + (\text{charge per mile} \times \text{number of miles}) \]
According to the problem, the pick-up fee is $4, and the charge per mile is $4.25. The total fare, before tip, is $110.25. Therefore, we can set up the equation as follows:
\[ 110.25 = 4 + 4.25x \]
To solve for \( x \), we first subtract the pick-up fee from both sides of the equation:
\[ 110.25 - 4 = 4.25x \]
\[ 106.25 = 4.25x \]
Next, we divide both sides by 4.25 to solve for \( x \):
\[ x = \frac{106.25}{4.25} \]
Calculating the division gives:
\[ x = 25 \]
Thus, the number of miles Arun traveled in the taxi is
\[ \boxed{25} \]
To verify, we can plug \( x \) back into the original fare calculation:
\[ \text{Total fare} = 4 + 4.25 \times 25 = 4 + 106.25 = 110.25 \]
This confirms our answer is correct. The taxi ride was indeed
\[ \boxed{25} \text{ miles.} \]
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