Proton is made of 3 quarks of each spin 1/2, but the total spin of proton also 1/2, how it is possible? Proton is made of 3 quarks $uud$ each has a spin of 1/2, so if you follow the constitution the total spin of the proton must be (3/2), but the spin of the proton is still 1/2. How it is possible?
But charge of quarks and proton it still follows the conservation (+2/3,+2/3,-1/3 = +1),
 A: See, the thing is that spin is actually a vector — it has also a direction. When considering such vector in quantum mechanics, 2 observables describe it completely: its norm ($S$) and projection on one of the axis (usually, $S_z$). For a spin-$\frac12$ particle the norm is $S=\frac12$ and $S_z = \pm \frac12$.
Then, for a system of 3 spin-$\frac12$ particles, the possible values for the sum of spins are $\frac12$ and $\frac32$.
So the quarks can take positive and negative values of spin. The proton has spin norm $1/2$ because it has two $+1/2$ spin and one $-1/2$ spin quarks, or two $-1/2$ spin and one $1/2$, because this is the same value of the norm, but different projection.
As to what is the particle with spin $\frac32$ — it is the so-called $\Delta_0$.
A: It's not as easy as Andrii Magalich explained - actually, why the proton has spin 1/2 isn't fully explained. See https://en.wikipedia.org/wiki/Proton for example for further reference, or https://physicsworld.com/a/the-spin-of-a-proton/ for a good overview.
In fact, the valence quarks (uud) only account for a fraction of the total proton spin. Probably the rest can be attributed to a combination of gluons and sea quarks.
