Step 1: Set the alphabetical order.
The letters of RANKS in dictionary order are $A<K<N<R<S.$ Rank means the position of the word when all $5!$ arrangements are listed alphabetically.
Step 2: Count words starting before $R$.
The first letter of our word is $R$. Letters that come before $R$ are $A,K,N$ (three of them). For each, the remaining $4$ letters can be arranged in $4!=24$ ways. That is $3\times24=72$ words first.
Step 3: Fix $R$, look at second letter.
Now first letter is $R$; remaining letters are $A,N,K,S$. Our second letter is $A$, the smallest, so no word comes before it here. Add $0.$
Step 4: Fix $RA$, look at third letter.
Remaining letters now are $N,K,S$. Our third letter is $N$. Letters before $N$ here: only $K$. For that one choice the rest arrange in $2!=2$ ways. Add $1\times2=2.$
Step 5: Fix $RAN$, fourth letter.
Remaining letters are $K,S$. Our fourth letter is $K$, the smaller one, so nothing comes before it. Add $0.$ The last letter $S$ is forced.
Step 6: Add and shift by one.
Words before RANKS: $72+0+2+0=74.$ The rank of RANKS itself is $74+1=75.$ \[ \boxed{75} \]