# 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.

Reveal Answer### Answer

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.

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