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We examine the coding problem of base layer (BL) in fine granularity scalable videos and rate adaptation of enhancement layer (EL) during streaming in order to support constant quality streaming. The problem arises from the facts that different frames of BL often exhibit significant quality variation in the usual FGS BL encoding methods, and consequently the actual EL Rate-Distortion (R-D) curves of different frames present significant difference. Under this condition it would be computationally exhaustive to scale the EL to flatten out the fluctuating BL quality and to attain constant quality. We propose, in this paper, an accurate constant quality encoding method of BL, under which we investigate the similarity of actual EL R-D curves, and then a simple but effective weight-based EL rate allocation algorithm is introduced. Experiments show, compared with default methods, our fast and simple methods not only provide constant-quality streaming, but also decrease the average MSE of videos.