# Marlene rides her bycicle to her friend Jon's house and returns home by the same route. Marlene rides her bike at constant speeds of 6mph on level grond, 4mph when going uphill and 12mph when going downhill. If her total riding time was 2 hours, how far was John's house? (note: Marlene does travel uphill, downhill, and on even ground)

14 years ago

8 years ago

## 4.5

6 years ago

## 22mph

11 months ago

## To solve this problem, we need to break down Marlene's journey into different segments based on her speed. Let's say she travels on level ground for x miles, uphill for y miles, and downhill for z miles.

Here's how we can approach the problem:

1. Calculate the time Marlene spends riding on level ground:

Time = Distance / Speed

The time spent on level ground is x / 6 (since her speed on level ground is 6mph).

2. Calculate the time Marlene spends going uphill:

Time = Distance / Speed

The time spent going uphill is y / 4 (since her speed uphill is 4mph).

3. Calculate the time Marlene spends going downhill:

Time = Distance / Speed

The time spent going downhill is z / 12 (since her speed downhill is 12mph).

4. Since her total riding time is 2 hours, we can write the equation:

x / 6 + y / 4 + z / 12 = 2

Now we need to find the values of x, y, and z. We can solve the equation using algebra.

Multiply the entire equation by the least common multiple (LCM) of 6, 4, and 12, which is 24.

24(x / 6) + 24(y / 4) + 24(z / 12) = 24(2)

4x + 6y + 2z = 48

Now we need to find a combination of x, y, and z that satisfies this equation and fits the problem's conditions. Since Marlene rides from her house to Jon's house and back, the distances traveled in each direction should be the same. Therefore, x + y + z is the total distance.

We also know that the total distance is equal to twice the distance from Marlene's house to Jon's house (round trip). So, x + y + z = 2d, where d is the distance from Marlene's house to Jon's house.

We can solve the system of equations simultaneously:

4x + 6y + 2z = 48 (Equation 1)

x + y + z = 2d (Equation 2)

We need another equation to eliminate one of the variables. In this case, let's use the equation specifying the relationship between the distances traveled at different speeds:

x + y + z = 2d

We can substitute the values for x, y, and z into this equation:

(6x / 6) + (4y / 4) + (12z / 12) = 2d

Simplifying,

x + y + z = 2d

Now we have two equations:

x + y + z = 2d (Equation 3)

4x + 6y + 2z = 48 (Equation 1)

We can solve this system of equations to find the values of x, y, z, and d.

This is a linear system of equations, and we can solve it using a variety of methods, such as substitution or elimination. However, to simplify the solution process, we will use the method of substitution.

From Equation 3, we can express x as a function of y and z:

x = 2d - y - z

Substitute this expression for x into Equation 1:

4(2d - y - z) + 6y + 2z = 48

8d - 4y - 4z + 6y + 2z = 48

8d + 2y - 2z = 48

Now we have two equations in terms of y, z, and d:

8d + 2y - 2z = 48 (Equation 4)

x + y + z = 2d (Equation 3)

We still have two variables (y and z), so we need another equation. We can use the information given in the problem that Marlene rides her bike for 2 hours:

Time = Distance / Speed

Total time = (x / 6) + (y / 4) + (z / 12)

Substituting the values for x and d:

(2d - y - z) / 6 + (y / 4) + (z / 12) = 2

Multiply through by 12 to eliminate the fractions:

2(2d - y - z) + 3y + z = 24

Simplifying:

4d - 2y - 2z + 3y + z = 24

4d + y - z = 24 (Equation 5)

So now we have three equations:

8d + 2y - 2z = 48 (Equation 4)

4d + y - z = 24 (Equation 5)

x + y + z = 2d (Equation 3)

We can now solve these equations simultaneously to find the values of x, y, z, and d, which will allow us to determine the distance from Marlene's house to Jon's house.