2022 AMC 10B Problem 4

Start small to see how many gaps there are.

1. Small cases. The 1st hiccup is at $4{:}00$ exactly. The 2nd is $5$ seconds later (one gap). The 3rd is $10$ seconds later (two gaps). So the $n$th hiccup comes after $n - 1$ gaps.
2. The 700th hiccup comes after $700 - 1 = 699$ gaps, which is
$$699 \cdot 5 = 3495 \text{ seconds after } 4{:}00.$$
3. Change to minutes. $58 \cdot 60 = 3480$, and $3495 - 3480 = 15$. So $3495$ seconds is $58$ minutes and $15$ seconds.
4. The time is $15$ seconds after $4{:}58$.

The answer is $\boxed{\textbf{(A)}\ 15 \text{ seconds after } 4{:}58}$.

This is the "fence post" idea: $n$ posts in a row have only $n - 1$ gaps between them. The hiccups are the posts and the $5$-second waits are the gaps. The answer choices are only $5$ seconds apart, which is exactly one gap, so they are designed to catch the off-by-one mistake. Checking the 2nd and 3rd hiccup first makes the right count clear.

The first hiccup is at $4{:}00$ exactly. Between the first and the $700$th hiccup there are $700 - 1 = 699$ intervals of $5$ seconds each, so the $700$th hiccup happens
$$699 \cdot 5 = 3495 \text{ seconds}$$
after $4{:}00$.

Convert: $3495 = 58 \cdot 60 + 15$, so that is $58$ minutes and $15$ seconds. The time is $15$ seconds after $4{:}58$.

The answer is $\boxed{\textbf{(A)}\ 15 \text{ seconds after } 4{:}58}$.