2008 AIME II Problems/Problem 2

Problem

Rudolph bikes at a constant rate and stops for a five-minute break at the end of every mile. Jennifer bikes at a constant rate which is three-quarters the rate that Rudolph bikes, but Jennifer takes a five-minute break at the end of every two miles. Jennifer and Rudolph begin biking at the same time and arrive at the -mile mark at exactly the same time. How many minutes has it taken them?

Solution

Let Rudolf bike at a rate , so Jennifer bikes at the rate . Let the time both take be .

Then Rudolf stops times (because the rest after he reaches the finish does not count), losing a total of minutes, while Jennifer stops times, losing a total of minutes. The time Rudolf and Jennifer actually take biking is then respectively.

Using the formula , since both Jennifer and Rudolf bike miles,

$\begin{align}r &= \frac{50}{t-245}\\

\frac{3}{4}r &= \frac{50}{t-120}

\end{align}$ (Error compiling LaTeX. Unknown error_msg)

Substituting equation into equation and simplifying, we find

$\begin{align*}50 \cdot \frac{3}{4(t-245)} &= 50 \cdot \frac{1}{t-120}\\

\frac{1}{3}t &= \frac{245 \cdot 4}{3} - 120\\ t &= \boxed{620}\ \text{minutes}

\end{align*}$ (Error compiling LaTeX. Unknown error_msg)