I am having trouble getting the answer to the miles in math homework

Question

When Fritz drives to work his trip takes 30 ?minutes, but when he takes the train it takes 20 minutes. Find the distance Fritz travels to work if the train travels an average of 12 miles per hour faster than his driving. Assume that the train travels the same distance as the car.
Fritz travels _______ miles to work.
simplify

Answer 1

The train travels an average 12 miles/hr

We always need to be working in the same units, so lets convert the trains speed to minutes since the time traveled is in minutes.

 

12 miles / 60 min = 0.2 miles/min

 

We are given that the train travels the same distance as the car.

 

The speed of each vehicle multiplied by the time traveled will give distance traveled. Since we don't know the speed of the train or car, we need to solve for one of them first.

 

Let "v" = speed of the car

Train travels 0.2 mile/hr faster than car

 

Trains speed = v + 0.2mile/min

 

Multiple each by it's time and set equal.

 

(v)*(30 min) = (v + 0.2mile/min)*(20 min)

30v = 20v + 4

 

10v = 4

v = 4/10 = 0.4 mile/min is the cars average speed

 

0.4 mile/min * 30 min = 12 miles

 

Answer 2

t₁ = driving time = 30 min = 1/2 hour

 

t₂ = train ride time = 20 min = 1/3 hour

 

v = driving speed

v + 12 = train speed (12 miles/hour faster)

 

The distance d is the same by car or by train:

 

d = vt₁ = v*(1/2 hour)

 

d = (v + 12)t₂ = (v + 12)*(1/3 hour)

 

Set the two expressions for d equal to each other, solve for v, then substitute v into d = vt₁

 

v*(1/2 hour) = (v + 12)*(1/3 hour)

 

1.5v = v + 12

 

0.5v = 12

 

v = 24 miles/hour

 

d = vt₁ = (24 miles/hour)*(1/2 hour) = 12 miles