Question

Dante drove to the mountains last weekend. There was heavy traffic on the way there, and the trip took 10 hours. When Dante drove home, there was no traffic and the trip only took 7 hours. If his average rate was 18 miles per hour faster on the trip home, how far away does Dante live from the mountains?Do not do any rounding.

Answer 1

Dante lives 420 miles away from the mountains.

Step-by-step explanation:

To find the distance Dante lives from the mountains, we can use the formula: Distance = Rate x Time. Let's assume the rate during the trip to the mountains is r, and the time is 10 hours. So, the distance to the mountains is 10r. On the trip home, the rate is r + 18 (since it's 18 miles per hour faster) and the time is 7 hours. So, the distance from the mountains to Dante's home is 7(r + 18).

Since the distance to and from the mountains is the same, we can set up an equation: 10r = 7(r + 18). Now, we can solve for r:

10r = 7r + 126

3r = 126

r = 42

Now, we can find the distance by plugging in the value of r into either equation:

Distance = 10(42) = 420 miles

Answer 2

You know that:

- Last weekend his trip took 10 hours when he drove to the mountains.

- When he drove home, the trip took 7 hours.

- His average rate was 18 miles per hour faster on the trip home.

By definition, the distance can be calculated with this formula:

`d=rt`

Where "d" is distance, "r" is rate, and "t" is time.

Then, you can set up the following equation to represent his trip to the mountains ("d" is in miles):

`d=10r`

And you can set up the following equation to represent his trip home ("d" is in miles):

`d=7(r+18)`

To find the value of "r", you need to make both equations equal to each other and solve for "r". Then, you get:

`\begin{gathered} 10r=7(r+18) \\ 10r=7r+126 \\ 10r-7r=126 \\ \\ r=(126)/(3) \\ \\ r=42 \end{gathered}`

Knowing the value of "r", you can substitute it into the second equation:

`\begin{gathered} d=7\mleft(r+18\mright) \\ d=(7)\mleft(42+18\mright) \end{gathered}`

Finally, evaluating, you get (Remember that "d" is in miles)

`\begin{gathered} d=(7)(60) \\ d=420 \end{gathered}`

Therefore, the answer is:

`420\text{ }miles`