Laura drove to the mountains last weekend. There was heavy traffic on the way there, and the trip took

6
hours. When Laura drove home, there was no traffic and the trip only took
4
hours. If her average rate was
22
miles per hour faster on the trip home, how far away does Laura live from the mountains?

rate of first trip --- x mph

rate of return trip = x+22 mph

6x = 4(x+22)

solve for x , then evaluate 6x

Let's call the distance Laura lives from the mountains "x" miles.

On the trip there, Laura's average speed was (x miles) / (6 hours) = (x/6) miles per hour.

On the trip home, Laura's average speed was (x miles) / (4 hours) = (x/4) miles per hour.

According to the information given, her average rate was 22 miles per hour faster on the trip home. So, we can set up the equation:

(x/4) = (x/6) + 22

To solve for x, we can multiply the equation by the least common multiple (LCM) of the denominators, which is 12:

12*(x/4) = 12*((x/6) + 22)

Simplifying this equation, we get:

3x = 2x + 264

Subtracting 2x from both sides gives:

x = 264

Therefore, Laura lives 264 miles away from the mountains.

To find the distance that Laura lives from the mountains, we can use the formula:

Distance = Rate × Time

Let x be the rate at which Laura drove to the mountains, in miles per hour. Based on the given information, her average rate on the trip home was 22 miles per hour faster than when she drove to the mountains, so her rate on the trip home would be (x + 22) miles per hour.

We know that the trip to the mountains took 6 hours, so the distance can be calculated as:

Distance = x × 6

Similarly, the trip back from the mountains took 4 hours, so the distance can be calculated as:

Distance = (x + 22) × 4

Since the distance to the mountains is the same as the distance back from the mountains, we can equate the two expressions:

x × 6 = (x + 22) × 4

Now let's solve this equation for x:

6x = 4(x + 22)

Simplify the equation:

6x = 4x + 88

Subtract 4x from both sides:

2x = 88

Divide both sides by 2:

x = 44

So, Laura's rate on the trip to the mountains was 44 miles per hour. To find the distance, we can use the equation:

Distance = Rate × Time

Distance = 44 miles/hour × 6 hours = 264 miles

Therefore, Laura lives 264 miles away from the mountains.

264 miles