Question

Together, Sam and Alice can paddle a canoe across the bay in 15 minutes. When Alice paddles alone, it takes her 16 minutes longer than when Sam paddles alone. How long does it take each person to paddle across the bay alone?

Alice takes ..... minutes to paddle across the bay alone.

Sam takes ...... minutes to paddle across the bay alone.

Answer 1

Can we assume that the speed of Alice and Sam together is the the speed of Alice + the speed of Sam?

Answer 2

Recall:
Velocity = distanceTravelled/time
Time = distanceTravelled/Velocity
DistanceTravelled = time*velocity

Vs = Sam's Velocity
Va = Alices' Velocity
Ts = Sam's Time
Ta = Alices' Time
Db = Bay's distance

Assume:
Speed of Alice and Sam together = Vs+Va

Given:
15min = Db/(Vs+Va)
Ta = Ts+16min

Now we'll use the two equations given and substitute one of the variables from one equation into the variable of the other...
15min = Db/(Vs+Va)
> 15Vs + 15Va = Db
> 15Vs + 15Db/Ta = Db
> 15Vs + 15Db/(Ts+16) = Db
*multiply both sides by (Ts+16)*

> 15VsTs + 16*15Vs + 15Db = TsDb+16Db
*VsTs = Velocity*Time = Distance = Db*

> 15Db + 16*15Vs + 15Db = TsDb + 16Db
*group Db's*

> 14Db + 240Vs = TsDb
> 240Vs = TsDb - 14Db
> 240Vs = Db(Ts-14)
> 240Vs/Db = Ts-14

*Velocity/Distance = 1/Time*

> 240/Ts = Ts-14
> 240 = Ts*Ts - 14*Ts
> Ts*Ts - 14Ts - 240 = 0

*Solve for Ts using quadratic equation or F.O.I.L. method*
>(Ts - 24)(Ts + 10) = 0 :: Ts = 24, -10
We cannot have negative time! Sam's Time alone must then be 24minutes
This means that Ta = Ts + 16 so Alice's Time alone is 16+24 = 40minutes

Ta = 40min
Ts = 24min

Hrm, I actually wasn't expecting to run across a quadratic for this, but the result seems right.

Answer 3

Check our work by plugging the times back into the first equation.
15min = Db/(Vs+Va)
15Vs + 15Va = Db
15Db/Ts + 15Db/Ta = Db

*substitute times for the times found*
15Db/24 + 15Db/40 ?= Db
0.625Db + 0.375Db ?= Db
1.0Db ?= Db ... yes!

Answer 4

Amazing
thank you very much