Andi and Brandi cycle around a field, with velocities: 8 km/h and 4 km/h. What is the perimeter of the field?

Question

$Andi$ and $Brandi$ cycle around a field, starts at same position and at same time with same direction. $Andi$'s and $Brandi$'s velocity are $8$ km/h and $4$ km/h. After one lap, $Andi$ took a rest for $5$ minutes, then continue. After of $20$ minutes of cycling, $Andi$ overlap $Brandi$. What is the perimeter of the field?

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Attempt:

Without a rest, $Andi$ will overlap $Brandi$ in exactly 2 laps. Because of $Andi$'s rest, $Brandi$ can take $4/12 = 1/3$ km advantage. So after exactly 2 laps, $Andi$ has $1/3$ km left to overlap $Brandi$.

$Brandi$ needs $5$ minutes to travel $1/3$ km. At the same amount of time, $Andi$ travels $2/3$.

So after exactly 2 laps, $Andi$ can cycle for $5$ more minutes until he overlaps $Brandi$.

So $20$ minutes = $X + 5$ minutes. Where $X$ is the amount of time of $Andi$ cycle 2 laps. So $X=15$. So 2 laps is $8/4 = 2$ km. One lap is $1$ km.

Are there better methods?

Answer 1

When $A$ starts cycling again $B$ will have covered $\frac{1}{2}$ a lap plus $\frac{1}{3}$ of a kilometre.

In the next $20$ minutes, $A$ travels $\frac{4}{3}$km more than $B$.

Therefore half a lap is $1$km and each lap is $2$km.

Answer 2

Let $L$ be the length of a lap, in km.

When Andy starts his rest, he has traveled $L$. Then when Andy cycles again after his rest, he cycles for 20 minutes at 8 km per hour so he travels $8/3$ km. So Andy travels $L+8/3$ km total.

As Brian's speed is half of Andy's, he has traveled $L/2$ when Andy starts his rest. In the 5+20 minutes after that [the5 minutes while Andy is resting and Brian keeps on cycling and then the 20 minutes after that], Brian cycles at 4km/hour so he travels $1/3 + 4/3$ km. So Brian has traveled $L/2 + 1/3 + 4/3$ km total.

In total Andy has traveled one more lap than Brian though. So we get the equation

$$\underbrace{L+ \frac{8}{3}}_{\text{distance Andy traveled}} \ = \ \underbrace{\left(\frac{L}{2} + \frac{1}{3} + \frac{4}{3}\right)}_{\text{distance Brian traveled}} +L $$

giving $L=2$km.

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OP in your reasoning there is missing that Andy was ahead of Brian by $L/2$ when Andy started his rest. You implicitly solved the equation

$$\frac{8}{3}\ = \ \frac{1}{3} + \frac{4}{3} +L $$

which is how you got $L=1$.