2006 AMC 10A Problem 4

The hour and the minutes do not affect each other, so make each one as big as possible on its own.

1. The hour. Hours go from $1$ to $12$. The one-digit hours give digit sums $1$ to $9$. The two-digit hours give only $1 + 0 = 1$, $1 + 1 = 2$ and $1 + 2 = 3$. So the best hour is $9$, with digit sum $9$.
2. The minutes. Minutes go from $00$ to $59$. The first digit can be at most $5$ and the second at most $9$, and $59$ has both. Its digit sum is $5 + 9 = 14$.
3. Put them together. The display $9{:}59$ is a real time, and its digits add to $9 + 14 = 23$. No display can beat both parts at once, so $23$ is the largest.

The answer is $\boxed{\textbf{(E)}\ 23}$.

When a total is made of separate pieces that do not affect each other, make each piece as large as it can be, then check the best pieces can happen together. Be careful that "biggest number" is not the same as "biggest digit sum": $12$ is a later hour than $9$, but its digits add to only $3$.

The hour part and the minute part are independent, so maximize each on its own.

Hours run from $1$ to $12$. A one-digit hour contributes its own value, at most $9$; the two-digit hours $10$, $11$, $12$ contribute only $1$, $2$, $3$. So the best hour is $9$, contributing $9$.

Minutes run from $00$ to $59$. The tens digit is at most $5$ and the ones digit at most $9$, and $59$ achieves both, contributing $5 + 9 = 14$.

The display $9{:}59$ therefore has digit sum $9 + 14 = 23$, and nothing larger is possible.

The answer is $\boxed{\textbf{(E)}\ 23}$.