# Problem of the Day

Ted and Rali run one lap on a circular track. Ted runs in a circular path with a radius of , and Rali runs in a circular path with a radius of , where . Is a factor of ?

(1) Ted and Rali finish one lap in the same amount of time.

(2) Rali’s average speed is times greater than Ted’s, where is an integer.

C. Both statements together are sufficient but neither statement alone is sufficient.

See the Solution

### Solution

[latexpage]

Statement 1: Ted and Rali finish one lap in the same amount of time. Ted runs a total distance of $2\pi r$ and Rali runs a total distance of $2\pi(r+q)$. Let $S_T$ be Ted’s average speed and $S_R$ be Rali’s average speed. If they both complete one lap in the same amount of time, then $\displaystyle \frac{2\pi r}{S_T}=\frac{2\pi r +2\pi q}{S_R}$.

Rearranging:

$\displaystyle \frac{S_R}{S_T}=\frac{2\pi r +2\pi q}{2 \pi r}$

$\displaystyle \frac{S_R}{S_T}=1+\frac{q}{r}$ (Separating the right side out to two distinct fractions and simplifying).

This tells us that the ratio of their speeds is $\displaystyle 1+\frac{q}{r}$, but it doesn’t give us any information about $r$ being a factor of $q$. INSUFFICIENT.

Statement 2:Â Rali’s average speed is $x$ times greater than Ted’s, where $x$ is an integer, or $S_R=xS_T$. This tells us nothing about $q$ and $r$. Rali could run $x$ times faster than Ted regardless of the values of $q$ and $r$. INSUFFICIENT.

Statements 1&2: Statement 2 tells us that $\displaystyle \frac{S_R}{S_T}$ is an integer. Combining that with statement 1 tells us that $\displaystyle 1+\frac{q}{r}$ is an integer, which can only be true if $r$ is a factor of $q$. SUFFICIENT.