Gemma recently rode her bicycle to visit her friend who lives 6 miles away. On her way there, her average speed was 16 miles per hour faster than on her way home. If Gemma spent a total of 1 hour bicycling, find the two rates.

first speed --- x mph

return speed -- x+16 mph

6/x + 6/(x+16) = 1
times each term by x(x+16)
6(x+16) + 6x = x(x+16)
x^2 + 4x - 96 = 0
(x-8)(x+12) = 0
x = 8 or x is a negative

her first speed was 8 mph
her return speed was 24 mph

check:
6/8 + 6/24 = 1 , that's good!

Well, it sounds like Gemma really wanted to make a grand entrance at her friend's place! Let's see if we can find her rates using a little math.

Let's suppose Gemma's speed on her way there was x miles per hour. According to the information given, her speed on her way back home would be x - 16 miles per hour. We also know that the total time Gemma spent riding her bicycle was 1 hour.

Now, let's break down the distances traveled. On her way there, Gemma rode for a distance of 6 miles, and her speed was x miles per hour. So, the time it took her to reach her friend's place was 6/x hours.

On her way back home, Gemma rode the same distance of 6 miles, but her speed was x - 16 miles per hour. So, the time it took her to get back home was 6/(x-16) hours.

Since the total time Gemma spent riding was 1 hour, we can set up the following equation:

6/x + 6/(x-16) = 1

Now, let's solve this equation and find the two rates, x and x - 16. But remember, I'm here for the laughs, not the math!

Let's assume that Gemma's speed on her way to her friend's house is "x" miles per hour. This means her speed on her way back home is "x - 16" miles per hour.

We can use the formula: Time = Distance/Speed to solve this problem.

On her way to her friend's house, Gemma traveled a distance of 6 miles. So, the time taken for this part of the journey is:
Time = Distance/Speed = 6/x

On her way back home, Gemma traveled the same distance of 6 miles. So, the time taken for that part of the journey is:
Time = Distance/Speed = 6/(x-16)

Now, we know that Gemma spent a total of 1 hour riding her bicycle. Therefore, we can write the equation:
Total time = Time to friend's house + Time back home
1 = 6/x + 6/(x-16)

To solve this equation, we need to find a common denominator. Multiplying both sides by x(x-16), we get:
x(x-16) = 6(x-16) + 6x

Expanding and simplifying the equation:
x^2 - 16x = 6x - 96 + 6x
x^2 - 16x = 12x - 96

Moving all terms to one side:
x^2 - 28x + 96 = 0

Now, we can solve the quadratic equation by factoring or using the quadratic formula. Let's solve it by factoring:
(x - 4)(x - 24) = 0

This gives us two possible solutions for x:
1. x - 4 = 0, which implies x = 4
2. x - 24 = 0, which implies x = 24

Since Gemma's speed cannot be negative, we discard the solution x = 24.

Therefore, Gemma's speed on her way to her friend's house is 4 miles per hour, and her speed on her way back home is 4 - 16 = -12 miles per hour.

However, a negative speed doesn't make sense in this context. So, we conclude that there is no valid solution to this problem.

To solve this problem, we can set up a system of equations based on the given information.

Let's assume that Gemma's speed on her way home is x miles per hour. Since her average speed on her way there was 16 miles per hour faster, her speed on her way there would be x + 16 miles per hour.

Now, let's calculate the time it took for Gemma to ride to her friend's house and the time it took for her to ride back home.

On her way there, Gemma traveled 6 miles at a speed of x + 16 miles per hour. Therefore, her time spent traveling to her friend's house would be:

time = distance / speed
t1 = 6 / (x + 16)

On her way back, Gemma traveled the same 6 miles, but at a speed of x miles per hour. Therefore, her time spent traveling back home would be:

time = distance / speed
t2 = 6 / x

Since Gemma spent a total of 1 hour bicycling, we can set up the equation:

t1 + t2 = 1

Now, substitute the expressions for t1 and t2:

6 / (x + 16) + 6 / x = 1

To solve this equation, we can multiply through by (x + 16)x, which gives us:

6x + 96 + 6(x + 16) = x(x + 16)

Simplifying the equation:

6x + 96 + 6x + 96 = x^2 + 16x

Combining like terms:

12x + 192 = x^2 + 16x

Moving all terms to one side of the equation:

x^2 + 4x - 192 = 0

Now we have a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. In this case, factoring is the most efficient method:

(x - 12)(x + 16) = 0

This gives us two possible values for x: x = 12 and x = -16. Since speed cannot be negative, we can discard x = -16.

Therefore, Gemma's speed on her way home is x = 12 miles per hour.

Using this value, her speed on her way there would be x + 16 = 12 + 16 = 28 miles per hour.

So, the two rates are 12 miles per hour and 28 miles per hour (on her way home and on her way there, respectively).