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

8 hours. When Austin drove home, there was no traffic and the trip only took
5 hours. If his average rate was
21 miles per hour faster on the trip home, how far away does Austin live from the mountains?
Do not do any rounding.

speed on first leg ---- x mph

speed on return trip -- x+21 mph

distance on first leg = 8x
distance for return leg = 5(x+21)
but the distances are equal, so

8x = 5(x+21)
8x = 5x + 105
3x=105
x = 35

distance = 8(35) = 280 miles

Just realized I did this same question a few weeks back

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Let's assume that the distance from Austin's home to the mountains is "d" miles.

On the way to the mountains, the trip took 8 hours, so his average speed was d/8 miles per hour.

On the way back home, the trip took 5 hours, and his average speed was (d/5) + 21 miles per hour.

Using the formula: speed = distance / time, we can set up the following equation:

d/8 = (d/5) + 21

To simplify the equation, we can multiply all the terms by 40 to eliminate the denominators:

5d = 8d + 840

Rearranging the terms, we have:

3d = 840

Dividing both sides of the equation by 3, we find the value of d:

d = 840 / 3

d = 280

Therefore, Austin lives 280 miles away from the mountains.

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

Distance = Rate × Time

Let's assume the distance from Austin's house to the mountains is represented by 'D' (in miles).

On the way there:
Time = 8 hours
Rate = unknown, let's call it 'R1'

So, for the journey to the mountains:
D = R1 × 8 ---- Equation 1

On the way back:
Time = 5 hours
Rate = R1 + 21 mph (since he was driving 21 mph faster)

So, for the journey back from the mountains:
D = (R1 + 21) × 5 ---- Equation 2

Now we need to solve these two equations to find the value of 'D'.

From Equation 1, we can rewrite it as:
R1 = D/8

Substituting this value of R1 into Equation 2:
D = (D/8 + 21) × 5

Let's solve for D:

D = (D/8) × 5 + 21 × 5
D = (5D/8) + 105
8D = 5D + 840
8D - 5D = 840
3D = 840
D = 840/3
D = 280

Therefore, Austin lives 280 miles away from the mountains.