It can be a little mind-bending trying to work out how much juice, concentrate or sugar syrup to add to the cider for carbonation.
However, FYI the following is how I do it. It is based on the idea that (Sugar in the cider) + (Sugar in the juice ) = (Sugar needed in the combined volumes of cider plus juice for carbonation). I don't know if this is the "official" way to solve the problem, but it works for me. Sounds a bit messy, but it can be expressed as a simple formula...
(Vc x Sc) + (Vj x Sj) = (Vc x Sf)+ (Vj x Sf), which solving for Vj = Vc x (Sf-Sc)/(Sj-Sf)
Using Vc = volume of cider (12L), Sc = g/L sugar in the cider (0g/L), Vj= volume of juice added, S j= g/L of sugar in the juice (118 g/L), and Sf = the final g/L of sugar needed for carbonation after the juice is added (9g/L).
So, for the original problem Vj = 12 x (9 - 0)/(118-9) = 108/109 = 0.99 litres, i.e. add 1 litre of juice to the cider to end up with a concentration of 9 g/L* of sugar (SG 1.005 for 2.5 vols of CO2).
* this figure is taken from the P&N table. Generally, at higher SGs there is reasonable agreement between published tables but wider variations for lower SGs, hence my preference for working with SG required for carbonation.