A new road that connects union town to springville is 4 1/3 miles long, what is the change in distance using the new road instead for the dirt road. Dirt road : 2 3/8 + 3 5/6

To find the change in distance by using the new road instead of the dirt road, we'll first calculate the total distance of the dirt road and then compare it to the new road.

### Step 1: Add the distances of the two segments of the dirt road.
- Segment 1: \( 2 \frac{3}{8} \)
- Segment 2: \( 3 \frac{5}{6} \)

First, convert these mixed fractions into improper fractions.

\[
2 \frac{3}{8} = \frac{2 \times 8 + 3}{8} = \frac{19}{8}
\]
\[
3 \frac{5}{6} = \frac{3 \times 6 + 5}{6} = \frac{23}{6}
\]

### Step 2: Find a common denominator for the fractions.
The least common denominator (LCD) of 8 and 6 is 24.

\[
\frac{19}{8} = \frac{19 \times 3}{8 \times 3} = \frac{57}{24}
\]
\[
\frac{23}{6} = \frac{23 \times 4}{6 \times 4} = \frac{92}{24}
\]

### Step 3: Add the two fractions.
\[
\frac{57}{24} + \frac{92}{24} = \frac{149}{24}
\]

Now, convert the improper fraction back to a mixed number:
\[
149 \div 24 = 6 \text{ remainder } 5
\]
So, \( \frac{149}{24} = 6 \frac{5}{24} \).

Thus, the total distance of the dirt road is \( 6 \frac{5}{24} \) miles.

### Step 4: Compare with the new road.
The length of the new road is \( 4 \frac{1}{3} \) miles. Convert this to an improper fraction:

\[
4 \frac{1}{3} = \frac{4 \times 3 + 1}{3} = \frac{13}{3} = \frac{104}{24} \quad (\text{common denominator with dirt road fraction})
\]

### Step 5: Calculate the change in distance.
Now, subtract the new road's distance from the dirt road's distance:

\[
6 \frac{5}{24} - 4 \frac{1}{3} = \frac{149}{24} - \frac{104}{24} = \frac{45}{24} = 1 \frac{21}{24} = 1 \frac{7}{8}
\]

### Final Answer:
The new road is \( 1 \frac{7}{8} \) miles shorter than the dirt road.

Would you like more details on any of these steps?

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**Related Questions:**
1. How do you convert improper fractions back to mixed numbers?
2. What are the steps for adding fractions with different denominators?
3. How would the process change if the dirt road had more segments?
4. What is the significance of finding a common denominator?
5. Can this process be used to compare times instead of distances?

**Tip:** Always convert mixed numbers into improper fractions before performing operations like addition or subtraction.

A new road that connects union town to springville is 4 1/3 miles long, what is the change in distance using the new road instead for the dirt road. Dirt road: 2 3/8 + 3 5/6

Solve the problem of comparing the length of a new road and a dirt road using mixed fractions. The total distance of the dirt road is calculated by adding fractions like 2 3/8 and 3 5/6. Suitable for Grades 5-7, this step-by-step guide explains key concepts such as converting mixed numbers to improper fractions and finding the least common denominator.

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