In an examination, the number of those who passed and failed were in the ratio 25:4. If three more had appeared and the number of those who failed was 3 less than earlier, the ratio of passers to failures became 21:3. The total number of students who initially appeared was?

Let the initial number of students who passed be $P_1$ and those who failed be $F_1$.
The initial ratio of passed to failed students is $25:4$.
So, we can write $P_1 = 25x$ and $F_1 = 4x$ for some constant $x$.
The initial total number of students is $T_1 = P_1 + F_1 = 25x + 4x = 29x$.

Three more students appeared, so the new total number of students is $T_2 = T_1 + 3 = 29x + 3$.
The number of students who failed was 3 less than earlier, so the new number of failed students is $F_2 = F_1 - 3 = 4x - 3$.
The new number of passed students is $P_2 = T_2 - F_2 = (29x + 3) - (4x - 3) = 29x + 3 - 4x + 3 = 25x + 6$.

The new ratio of passers to failures is $21:3$, which simplifies to $7:1$.
So, $\frac{P_2}{F_2} = \frac{7}{1}$.
Substitute the expressions for $P_2$ and $F_2$:

Multiply both sides by $(4x - 3)$:

Subtract $25x$ from both sides:

Add 21 to both sides:

Divide by 3:

The initial total number of students was $T_1 = 29x$.
Substitute the value of $x$:

The total number of students who initially appeared was 261.

The final answer is $\text{261}$.