Alright class, let's break down and solve this question step by step:
### Step 1: Interpret the Trail Lengths
We are given two trail lengths:
1. [tex]\( 1 \frac{3}{4} \)[/tex] miles
2. [tex]\( 2 \frac{1}{2} \)[/tex] miles
### Step 2: Convert Mixed Numbers to Improper Fractions or Decimals
To make the calculations easier, let's convert these mixed numbers to decimals:
1. [tex]\( 1 \frac{3}{4} \)[/tex] is equivalent to [tex]\( 1 + \frac{3}{4} \)[/tex] which is [tex]\( 1.75 \)[/tex] miles.
2. [tex]\( 2 \frac{1}{2} \)[/tex] is equivalent to [tex]\( 2 + \frac{1}{2} \)[/tex] which is [tex]\( 2.5 \)[/tex] miles.
### Step 3: Calculate the Total Length of All Trails
The total length of the trails is the sum of these two trail lengths:
[tex]\[ 1.75 \, \text{miles} + 2.5 \, \text{miles} = 4.25 \, \text{miles} \][/tex]
### Step 4: Verifying Chang's Statement
Chang says the longest ski trail is more than three times the length of the shortest ski trail.
To verify this, first, identify the longest and shortest trails:
- Longest ski trail: [tex]\( 2.5 \)[/tex] miles
- Shortest ski trail: [tex]\( 1.75 \)[/tex] miles
Next, calculate three times the length of the shortest ski trail:
[tex]\[ 3 \times 1.75 \, \text{miles} = 5.25 \, \text{miles} \][/tex]
Now, check if the longest ski trail is more than three times the shortest ski trail:
[tex]\[ 2.5 \, \text{miles} > 5.25 \, \text{miles} \][/tex]
Clearly, [tex]\( 2.5 \)[/tex] miles is not greater than [tex]\( 5.25 \)[/tex] miles. Therefore, Chang's statement is false.
### Summary of Results
- The total length of all trails is [tex]\( 4.25 \)[/tex] miles.
- Chang's statement that the longest ski trail is more than three times the length of the shortest ski trail is false.
By following these steps, we have systematically solved the problem and confirmed the key numerical results.
