Some kids went on a bike ride. The ride was six hours long and the kids traveled 54 miles. For part of the ride they biked at a speed of 7 miles per hour and for part of the ride they biked at a speed of 11 miles per hour. For how long did they bike at the first speed and for how long at the other speed?

Step-by-step explanation:

The total distance they travel is 54miles

And the total time taken for the travel is 6hours.

First part of the journey

They traveled at speed 7miles/hour

Let the time for this period be t

Given that, speed = distance / time

Then, distance = speed * time

Also, let the distance covered for this journey be y

Since, d = s*t

y=7t Equation 1

Part two of the journey

They travel at speed 11miles/hour

If the time they spend in the first journey is t and the total time for the journey is 6hours, then the time for the second part of the journey is (6-t) hours

Also, if for the first part of the journey he traveled a distance of y and the total distance for the journey is 54miles, then he traveled 54-y miles for the second part

Given that, speed = distance / time

Then, distance = speed * time

54-y=11 (6-t)

54-y=66-11t

11t-y=66-54

11t-y=12 Equation 2

Substitute equation 1 into 2. y=7t

11t-y=12

11t-7t=12

4t=12

t=3hours

Then the time for the first part of the journey is 3hours

The time for the second part is given as (6-t) = 6-3=3hours

Then, the time for the first part and second part are equal which is 3hours