Question

Tim and Rick both can run at speed vr and walk at speed vw, with vr>vw. They set off together on a journey of distance D. Rick walks half of the distance and runs the other half. Tim walks half of the time and runs the other half.

Part A
How long does it take Rick to cover the distance D?
Express the time taken by Rick in terms of vr, vw, and D.

Part B
Find Rick's average speed for covering the distance D.
Express Rick's average speed in terms of vr and vw.

Part C
How long does it take Tim to cover the distance?
Express the time taken by Tim in terms of vr, vw, and D.

Answer 1

Answers:

Part A: T = [ D/2 ] [ 1 / Vr + 1 / Vw) ]

Part B: Average speed = 2 × Vr × Vw / [ Vr + Vw ]

Parte C: T = 2 D / [Vr + Vw]

1) Variables:

D = total distance

Dr = run distance

Dw = walk distance

Vr = run speed

Vw = walk speed

T = total time

Tr = run time

Tw = walk time

2) Formulas:

i) velocity = distance / time

ii) distance = velocity × time

iii) time = distance / velocity

iv) Average speed = total distance / total time

3) General table:

person---------- run ------------ walk

RicK-------------- D/2------------- D/2

Tim -------------- T /2 ------------- T /2

4) Part A. How long does it take Rick to cover the distance D?

i) T = Tr + Tw

ii) Tr = Dr / Vr

iii) Dr = D / 2

⇒ Tr = [D/2] / Vr = D / [ 2Vr]

iv) Tw = [D/2] / Vw = D / [2Vw]

v) T = Tr + Tw = [D / 2] × [ 1 / Vr + 1 / Vw ]

T = [ D/2 ] [ 1 / Vr + 1 / Vw) ] ← answer

5) Part B. Find Rick's average speed for covering the distance D.

Average speed = D / T = D × 1 / T = D × [2 / D] × [Vr×Vw] / [Vr + Vw]

Average speed = 2 × Vr × Vw / [ Vr + Vw ]

5) Part C How long does it take Tim to cover the distance?

i) Tr = Dr / Vr = T / 2 ⇒ Dr = Vr × T /2

ii) Tw = Dw / Vw = T / 2 ⇒ Dw = Vw × T /2

iii) ⇒ D = Dr + Dw = Vr × T / 2 + Vw × T / 2

⇒ D = [Vr + Vw] × T / 2

iv) ⇒ T = 2 D / [Vr + Vw] ← answer.

Answer 2

So let's take by parts:
Part A: the total time is the sum of the time for each leg, and those times are half the distance divided by the speed. In formula that would be like this:

time = ( (D/2)/vw) + (D/2)/vr )

Part B: Average speed is total distance divided by total time.

average speed = D/time = D / ( (D/2)/vw) + (D/2)/vr ) = 1 / ( 1/(2*vw) + 1/(2*vr) ) = 1 / ( (2*vw + 2*vr) / (2*vr*2*vw) = 4*vr*vw / (2*vw + 2*vr)

Part C: time = D*2/(vw+vr)

If you had a D, asking for Tim's average speed, for that, the formula would be (vr+vw)/2