3. (10 points) The distance between $A$ and $B$ is 500 kilometers. Two people, Jia and Yi, start cycling from $A$ to $B$ at the same time. Jia rides 30 kilometers per day, and Yi rides 50 kilometers per day but rests every other day. At the end of the $\qquad$th day, the distance Yi has left to $B$ is twice the distance Jia has left to $B$.
【Analysis】If the number of days is even, Yi is equivalent to riding $50 \div 2=25$ (kilometers) per day. Let the distance from Yi to B at the end of the $n$-th day be twice the distance from Jia to B. According to the problem, we get the equation: $(500-30 n) \times 2=500-25 n$. Solving this, we get: $n=14.2857$. The number of days cannot be a decimal, so the number of days must be odd. Let the distance from Yi to B at the end of the $(n+1)$-th day be twice the distance from Jia to B, and then set up the equation to solve it.
【Solution】Solution: If the number of days is even,
Yi is equivalent to riding $50 \div 2=25$ (kilometers) per day,
Let the distance from Yi to B at the end of the $n$-th day be twice the distance from Jia to B. According to the problem, we get the equation:
$$
(500-30 n) \times 2=500-25 n
$$

Solving this, we get: $n=14.2857$,
The number of days cannot be a decimal, so the number of days must be odd.
Let the distance from Yi to B at the end of the $(n+1)$-th day be twice the distance from Jia to B. According to the problem, we get the equation:
$$
500-25 n-50=[500-30(n+1)] \times 2
$$

Solving this, we get: $n=14$,
$$
n+1=15 \text { (days); }
$$

Answer: At the end of the 15th day, the distance from Yi to B is twice the distance from Jia to B.