Jen recently rode her bicycle to visit her friend who lives 12 miles away. On her way there, her average speed was 7 miles per hour faster than on her way home. If Jen spent a total of 1 hour bicycling, find the two rates.

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Answer:

Rate while going to her friend's home = 28 mph
Rate while her way back home = 21 mph

Step-by-step explanation:

Let the average speed of Jen while her way back home be x mph. 

So, her average speed while going to her friend's home = (x + 7)mph

Now, distance = 12 km

So, time taken while going to her friend's home = Distance/speed = 12/(x + 7) hours

Time taken while coming her way back home = Distance/speed = 12/x hours

Now, total time taken = 1 hour

Thus, 

12/x + 12/(x + 7) = 1

(12x + 84 + 12x)/x(x + 7) = 1

24x + 84 = x(x + 7) 

24x + 84 = x² + 7x

x² - 17x - 84 = 0

Putting -17x = 4x - 21x, 

x² + 4x - 21x - 84 = 0

x(x + 4) - 21(x + 4) = 0

(x + 4)(x - 21) = 0

So, x = -4 or x = 21

Since speed cannot be negative, so x = 21 mph

Thus, 

Rate while going to her friend's home = x + 7 = 21 + 7 = 28 mph

Rate while her way back home = x = 21 mph

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