Balancing chemical systems - searching and sorting preliminary information
From your diagram, $R_G$ is the recycled gas - the hydrogen that has not been consumed plus the methane that has been produced. $R_L$ is the recycled liquid - the unreacted toluene. Both go back into the mixer with fresh toluene added at the rate given and fresh hydrogen added to bring the mole ratio of hydrogen to toluene (and hydrogen to methane) to what is given.

$R_L$ is easier. The problem tells you that 80% of the toluene is converted into benzene and 8% of the toluene is converted into biphenyl. What percentage of toluene is left over? The original feed flow of toluene is 3450 lb/hr, but the amount of toluene entering the gross feed to the reactor is higher (includes $R_L$). 

$$G_{\text{tol}}=F+R_L$$
$$R_L = 0.12\times G_{\text{tol}}$$

For $R_G$, you need to know how much hydrogen you have going into the mixer. You are told that the mole ratio of hydrogen to toluene in the gross feed is 4:1. Now that you have $R_L$, you can figure out the total toluene in the gross feed, and then do the stoichiometry calculation to get to total hydrogen in the gross feed. Since you have the balanced equations, you can determine how much hydrogen is consumed. If 80% of the toluene is consumed in the generation of benzene, how much hydrogen is consumed? How much methane is produced? Do the same with the conversion of toluene to biphenyl.

Once you are able to obtain the molar flow rates of all molecules out from the reactor, you should be able to calculate the flow rate of the gas exiting from the separator. This gas stream (Let's call it $K$) contains only $\ce{CH4}$ and $\ce{H2}$ of mol fractions $y$ and $1-y$ respectively. The mol fractions can be calculated considering that all $\ce{CH4}$ and $\ce{H2}$ that exists the reactor leaves the separator in $K$.

The molar composition of the $R_G$ and $P$ streams that leave $K$ is the same. This means that you are now able to use a molar flow rate balance on the mixing point between $M$, $R_G$ and the stream to the mixer (Lets's call this $Y$).

$$Y = M + R_G$$

By doing a balance on methane, you can find $R_G$.

$$(0.5)*(Y) = (1-y)(R_G) $$

Since you know $Y$ and $y$, you will then be able to find $R_G$.