Trying to find the time traveling by bike

Question

After cycling at a steady rate for 60 miles, Cecil had a flat tire. He walked 8 miles to a repair shop. Cecil's cycling rate was 3 times faster than this walking rate. If the time spent cycling and walking was 7 hours, at what rate was 'Cecil riding his bike?

Answer 1

Since Cecil's biking speed is given as 3 times his walking speed, 

let r = his biking speed and let r/3 = his walking speed.

 

This is a d = rt problem, but it's trickier than some since there are two different distances and speeds, adding up to a total time of 7 hrs.

 

The adjusted formula for t is t = d/r

 

The expression for the time spent biking, then, is 60/r

The expression for the time spent walking is 8 over r/3, which can be re-written as 24/r

 

The sum of the time spent biking and walking is 7 hrs, so

 

60/r + 24/r = 7

60 + 24 = 7r (multiplying through by r)

84 = 7r  (simplifying left side)

12 = r (dividing both sides by 7)

 

So Cecil biked at 12 mph.  (ANSWER)

 

Check this by plugging back into the basic d = rt formula:

 

If he rode 60 miles at 12 mph, it would have taken 5 hours

 

If he walked 8 miles at 1/3 that speed, i.e., 4 mph, that would have taken 2 hours.

 

5 hours + 2 hours = 7 hours, so our answer checks.

 

 

 

Answer 2

Wow biking 60 miles and walking 8 miles, that would tire me out.

 

Cecil travelled a total of 68 miles over 7 hours and you have 2 equations for this.

 

D(1) = R(1) * T(1)

 

60 = R(1) * T(1)

8   = R(2) * T(2)

 

Now we know that T(1) + T(2) = 7 and R(1) = 3 * R(2) so let's do some rearranging and substitutions.

 

First solve for T in both equations:

 

T(1) = 60 / R(1)

T(2) = 8 / R(2)

 

We can add these 2 equations

 

T(1) + T(2) = 60 / R(1) + 8 / R(2)

 

Let's substitute what we know about T(1) + T(2)

 

7 = 60 /R(1) + 8/R(2)

 

And now substitute what we know about R(1)

 

7 = 60/(3R(2)) + 8/R(2)

 

We need common denominators so multiply the second denominator by 3 and the numerator also

 

7 = 60/3R(2) + 24/3R(2)

 

Let's add 60 and 24 and then divide by 3 and we are left with

 

7 = 28/R(2)

 

Solving for R(2) leaves us with 

 

R(2) = 28/7  (hope you have your algebra skills in order)

 

So finally R(2) = 4 and that would be 4 MPH that the Cecil walked (quite a trot) and he biked 3 times that so 12 miles per hour.

 

Go back and check our work

 

Cecil walked 8 miles at 4 MPH so it took 2 hours

Cecil cycled 60 miles at 12 MPH so it took 5 hours

 

That is a total of 7 hours, so this answer looks good to me, hope it helped you!

 

All the best

 

Lori

 

 

 

 

 

 

Answer 3

distance=rate*time

60=(3r)(t)

8=r(7-t)

60=(3r)(t) means 60/3r=t, or 20/r=t

substitute

8=r(7-[20/r])

8=7r-20

8+20=7r

28=7r

7r=28

r=28/7=4 mph walking and 3*4=12 mph riding his bike