To determine which equation represents the proportional relationship between the distance [tex]\( d \)[/tex] and the time [tex]\( t \)[/tex], given that a train is traveling at a constant speed and has traveled 67.5 miles in [tex]\( 1 \frac{1}{2} \)[/tex] hours, we need to calculate the speed of the train and use this to form the equation.
### Step-by-Step Solution:
1. Convert Mixed Number to Improper Fraction:
The time given is [tex]\( 1 \frac{1}{2} \)[/tex] hours. First, let’s convert this mixed number into an improper fraction:
[tex]\[
1 \frac{1}{2} = 1 + \frac{1}{2} = \frac{2}{2} + \frac{1}{2} = \frac{3}{2} \text{ hours}
\][/tex]
2. Calculate Speed:
The train traveled 67.5 miles in [tex]\( \frac{3}{2} \)[/tex] hours. The speed [tex]\( s \)[/tex] of the train can be calculated using the formula:
[tex]\[
s = \frac{\text{distance}}{\text{time}} = \frac{67.5 \text{ miles}}{\frac{3}{2} \text{ hours}}
\][/tex]
Instead of dividing by a fraction, we multiply by its reciprocal:
[tex]\[
s = 67.5 \div \frac{3}{2} = 67.5 \times \frac{2}{3}
\][/tex]
3. Perform the Multiplication:
[tex]\[
s = 67.5 \times \frac{2}{3} = 67.5 \times 0.6667 = 45 \text{ miles per hour}
\][/tex]
Therefore, the train's speed is 45 miles per hour.
4. Form the Equation:
Since the speed is 45 miles per hour, the proportional relationship between the distance [tex]\( d \)[/tex] and the time [tex]\( t \)[/tex] can be expressed as:
[tex]\[
d = 45t
\][/tex]
### Conclusion:
Thus, the equation that correctly shows the proportional relationship between the distance [tex]\( d \)[/tex] and the time [tex]\( t \)[/tex] that the train has traveled is:
[tex]\[
d = 45t
\][/tex]
The correct answer is:
[tex]\[
d = 45t
\][/tex]
