Math  /  Algebra

QuestionFelipe drove to the mountains last weekend. There was heavy traffic on the way there, and the trip took 8 hours. When Felipe drove home, there was no traffic and the trip only took 6 hours. If his average rate was 16 miles per hour faster on the trip home, how far away does Felipe live from the mountains?
Do not do any rounding.

Studdy Solution

STEP 1

1. The trip to the mountains took 8 hours.
2. The trip back home took 6 hours.
3. Felipe's average speed on the way back was 16 miles per hour faster than on the way to the mountains.
4. We are trying to find the distance from Felipe's home to the mountains.

STEP 2

1. Define variables for the speeds and distance.
2. Write equations for the distance based on the time and speed.
3. Use the relationship between the speeds to create an equation.
4. Solve the system of equations to find the distance.

STEP 3

Define variables for the speeds and distance.
Let d d be the distance from Felipe's home to the mountains. Let r r be the average speed on the way to the mountains. Then, the average speed on the way back is r+16 r + 16 miles per hour.

STEP 4

Write equations for the distance based on the time and speed.
The distance to the mountains is given by the formula: d=r×8 d = r \times 8
The distance back home is given by the formula: d=(r+16)×6 d = (r + 16) \times 6

STEP 5

Use the relationship between the speeds to create an equation.
Since both expressions represent the same distance d d , we can set them equal to each other: r×8=(r+16)×6 r \times 8 = (r + 16) \times 6

STEP 6

Solve the system of equations to find the distance.
First, expand the equation: 8r=6r+96 8r = 6r + 96
Subtract 6r 6r from both sides to isolate terms with r r : 2r=96 2r = 96
Divide both sides by 2 to solve for r r : r=48 r = 48
Now, substitute r=48 r = 48 back into the equation for distance: d=r×8=48×8 d = r \times 8 = 48 \times 8
Calculate the distance: d=384 d = 384
The distance from Felipe's home to the mountains is: 384 \boxed{384}

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