Final answer:
The change in distance when using the new road instead of the dirt roads is -16 25/288 miles.
Step-by-step explanation:
To find the change in distance when using the new road instead of the dirt roads, we need to subtract the total distance of the dirt roads from the length of the new road. The total distance of the dirt roads is 2 3/8 + 3 5/6 miles. To subtract mixed numbers, we need to convert them to improper fractions.
2 3/8 = 19/8 and 3 5/6 = 23/6. Now we can subtract: 4 1/3 - (19/8 + 23/6).
To subtract fractions, we need a common denominator. The least common multiple (LCM) of 8 and 6 is 24.
Converting the fractions to have a denominator of 24, we have: (4*(24/3) - ((19*(3/3)) + (23*(4/4))))/24.
Simplifying the expression, we get: (96/3 - (57/3 + 92/4))/24 = (96/3 - 229/12)/24.
Now, finding the difference: (96/3 - 229/12)/24 = (32 - 229/12)/24 = (-197/12)/24 = -197/288.
Therefore, the change in distance when using the new road instead of the dirt roads is -197/288 miles, which can be written as a mixed number as -16 25/288 miles.