A train traveled for three hours. In the first hour the train traveled \(86\) miles, which was \(25\%\) farther than

To determine the total distance traveled by the train, we need to break down the problem hour by hour.

1. First Hour:
- The train traveled 86 miles in the first hour.

2. Second Hour:
- We know that 86 miles was 25% farther than the distance traveled in the second hour.
- Let \( d \) be the distance traveled in the second hour.
- Thus, 86 miles = \( d + 0.25d \).
- Therefore, \( 86 = 1.25d \).
- Solving for \( d \):
\[
d = \frac{86}{1.25} = 68.8 \text{ miles}
\]

3. Third Hour:
- The train traveled at a speed of 120 miles per hour for 20 minutes.
- 20 minutes is \(\frac{20}{60} = \frac{1}{3}\) of an hour.
- Therefore, the distance traveled in the third hour is:
\[
\text{Distance} = 120 \text{ miles per hour} \times \frac{1}{3} \text{ hour} = 40 \text{ miles}
\]

4. Total Distance Traveled:
- Add the distances from each hour:
\[
\text{Total distance} = 86 \text{ miles} + 68.8 \text{ miles} + 40 \text{ miles}
\]
\[
\text{Total distance} = 194.8 \text{ miles}
\]

Thus, the total distance that the train traveled is \( \boxed{194.8} \).