In $\triangle{XYZ}$, $\angle{Z}=90^{\circ}$, $\cos{Y}=\frac{7}{25}$, and $XY=25$.  What is $\tan{X}$?
Since $\angle{Z}$ is the right angle, our triangle is shown below:

[asy]
draw((0,0)--(7,0)--(7,24)--cycle,black+linewidth(1));
draw(rightanglemark((0,0),(7,0),(7,24),30),black+linewidth(1));
label("$X$",(7,24),E);
label("$Y$",(0,0),W);
label("$Z$",(7,0),E);
label("25",(0,0)--(7,24),NW);
[/asy]

Since $\cos{Y}=\frac{7}{25}$, we have $\cos{Y}=\frac{YZ}{XY}=\frac{YZ}{25}=\frac{7}{25}$.  From this, we can see that $YZ=25\cdot\frac{7}{25}=7$.  Then, from the Pythagorean Theorem, $XZ=\sqrt{XY^2-YZ^2}=\sqrt{25^2-7^2}=\sqrt{576}=24$.  Finally, we have $\tan{X}=\frac{YZ}{XZ}=\boxed{\frac{7}{24}}$.