How to prove $Q_u = -2Q_d$ from $SU(5)$? I'm a beginner, and I'm trying to figure out how to prove that charge of Up quark is equal to 2 times the charge of down quark from the 10 representation of $SU(5)$. Please help.
 A: You are giving very little information on what you know, your conventions, etc...
With the conventions of T P Cheng & L F Li, 
$$
\lambda_0=\frac{1}{\sqrt{15}}\operatorname{diag}(2,2,2,-3,-3)=-\frac{6}{\sqrt{15}} \frac{Y}{2},\\
\lambda_3= \operatorname{diag}(0,0,0,1,-1)=2T_3,\implies\\
Q= \operatorname{diag}(-1/3,-1/3,-1/3,1,0),
$$
by Gell-Mann–Nishijima. 
In these conventions, 

You then simply read off the charges of your 5 by left multiplication by Q; of its conjugate by left multiplication by -Q; and on the antisymmetric rep 10 by $Q\chi + \chi Q$. You should be able to confirm all charges.  The easiest entries to see the magic in are the 4th and 5th columns, and 4th and 5th rows. 
Of course, if you already have the Y and T3 numbers of each multiplet, prepackaged, you don't need to do anything like that: you simply apply Gell-Mann–Nishijima.  
For the isosinglets, where Q=Y/2, in the upper 3 components of the 5 * , and the upper 3×3 block of the 10, you simply see the - sign and the 2 relationship without even specifying specific values for Y, simply knowing its form, the degeneracy of its  upper 3 components' piece so as to commute with color.
