# Find common arrival time, given a system of equations

Ginny and Jenna are 20 miles from home. They have one pair of roller skates. Jenna walks 4mph and skates 9mph. Ginny walks 3 mph and skates 8mph. They start for home at the same time.

First, Ginny has the skates and Jenna walks. Ginny skates for a while, then takes the roller skates off and starts walking.When Jenna reaches the roller blades, she puts them on and starts skating home.

If they both start at 4:00 and arrive home at the same time, what time is it when they get home?

My solution,

I assumed that the time that Ginny skates is $a$ hours and walks for $b$ hours. And hence Jenna skates for $b$ hours and walks for $a$ hours. And since total distance covered is 20 for both, I got the following 2 equations.

\begin{align} 8a + 3b &= 20 \\ 4a + 9b &= 20 \end{align}

I solved this system of equations by elimination to get $b = \dfrac{4}{3}$ and $a = 2$, and $a + b = \dfrac{10}{3}$. This doesn't check out with the required solution which is $4$ and arriving at $8$ pm.

I have checked the simultaneous equation, so I have probably made a logic error. Any ideas where I went wrong. Thanks.

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You’ve got Jenna walking for $a$ hours, but in fact she must walk longer than that: after $a$ hours she still has to cover the distance to the point at which Ginny took off the skates. –  Brian M. Scott Mar 4 '12 at 13:05

Suppose, as you did, that Ginny skates for $a$ hours and walks for $b$ hours; that gives us one relationship between $a$ and $b$, namely, $$8a+3b=20\;.\tag{1}$$
After $a$ hours Ginny has covered $8a$ miles, and Jenna, walking at $4$ mph, has covered $4a$ miles. It will take Jenna another $a$ hours to reach the skates. Ginny covered the remaining distance in $b$ hours, walking at $3$ mph, so the remaining distance is $3b$ miles. Skating at $9$ mph, Jenna will need $\frac{3b}9=\frac{b}3$ hours to cover this distance. Thus, Jenna’s total elapsed time must be $2a+\frac{b}3$. But we know that they took the same total amount of time, so $$2a+\frac{b}3=a+b\;,$$ or $$a=\frac23b\;;\tag{2}$$ this gives us a second relationship between $a$ and $b$. Now just solve the system consisting of equations $(1)$ and $(2)$.