Find the ratio of Alan and Jenny's speeds

[mathgeek7365]

Alan is a jogger that leaves the top of a hill heading for the bottom of the hill. At the same time, Jenny, a biker, heads up the hill attempting to reach the top. They each proceed at a steady rate. After passing each other, Jenny takes 81 times as long to reach the top as Alan takes to reach the bottom. What is the ratio of their speeds?

D₁ is Alan's first distance, D₂ is Alan's second distance, T₁ is Alan's first time, T₂ is Alan's second time, R is Alan's rate, d₁ is Jenny's first distance, d₂ is Jenny's second distance, t₁ is Jenny's first time, t₂ is Jenny's second time, and r is Jenny's rate. Note that "first distance, second distance, first time, and second time" is before or after they meet. For example, d₂ would be Jenny's distance traveled after she passes Alan.

What we have done so far is writing different equations based on the information given/known. We know that d=rt, and can be rearranged to be r=d/t or t=d/r. These are the equations we have so far:
D₁= R(T₁), d₁= r(t₁)
D₂= R(T₂), d₂= r(81t₁)

We also rearranged some of the equations above, so:
81T₂= d₂/r, T₁= D₁/R, and t₁= d₁/r

This is all we have. Is this right so far? If so, how do we finish the problem? If not, what do we need to do differently?

 

[Euge]

Hi mathgeek7365,

Let $h$ be the vertical distance between the top and bottom of the hill. In time $t$, Alan has traveled a distance equal to $h - Rt$; in time $t$, Jenny has traveled a distance equal to $rt$. Alan and Jenny meet when $h - Rt = rt$. Solving for $t$, we find that they meet when

$$t = \frac{h}{r + R}$$

Let $t_0$ represent that value of $t$. If $t_A$ is the time Alan takes to reach the bottom from the top of the hill, then his vertical distance from the ground is zero, i.e., $h - Rt_A = 0$. Solving for $t_A$ yields $t_A = \frac{h}{R}$. If $t_J$ is the time Jenny takes to reach the top from the bottom of the hill, then her vertical distance from the ground is $h$, i.e., $rt_J = h$. So $t_J = \frac{h}{r}$. Now since after they meet, it takes Jenny $81$ times as long to reach the top as it takes Alan to reach the bottom, then

$$t_J - t_0 = 81(t_A - t_0)$$

$$\frac{h}{r} - \frac{h}{r + R} = 81\left(\frac{h}{R} - \frac{h}{r + R}\right)$$

$$\frac{hR}{r(r + R)} = 81\frac{hr}{R(r + R)}$$

$$\frac{R}{r} = \frac{81r}{R}$$

Cross multiplying results in $R^2 = 81r^2$; taking square roots on both sides gives $R = 9 r$. Hence

$$R : r = 9 : 1$$

 

[MarkFL]

Another way to approach this is to let:

$$d_A$$ = the distance traveled by Alan after they meet

$$d_J$$ = the distance traveled by Jenny after they meet

Thus:

$$d_A=r_At$$

$$d_J=r_J(81t)$$

From this, we determine:

$$\frac{r_A}{r_J}=81\frac{d_A}{d_J}$$

Now, since the distance traveled by Alan before they meet is equal to the distance Jenny travels after they meet, and likewise the distance traveled by Jenny before they meet is equal to the distance Alan travels after they meet we also have:

$$\frac{d_A}{d_J}=\frac{r_J}{r_A}$$

Thus:

$$\frac{r_A}{r_J}=81\frac{r_J}{r_A}\implies \frac{r_A}{r_J}=9$$

And so we conclude:

$$r_A:r_J=9:1$$