Compression standards have been used to reduce the cost of image storage and transmission for decades. In recent years, learned image compression methods have been proposed and achieved compelling performance to the traditional standards. However, in these methods, a set of different networks are used for various compression rates, resulting in a high cost in model storage and training. Although some variable-rate approaches have been proposed to reduce the cost by using a single network, most of them brought some performance degradation when applying fine rate control. To enable variable-rate control without sacrificing the performance, we propose an efficient Interpolation Variable-Rate (IVR) network, by introducing a handy Interpolation Channel Attention (InterpCA) module in the compression network. With the use of two hyperparameters for rate control and linear interpolation, the InterpCA achieves a fine PSNR interval of 0.001 dB and a fine rate interval of 0.0001 Bits-Per-Pixel (BPP) with 9000 rates in the IVR network. Experimental results demonstrate that the IVR network is the first variable-rate learned method that outperforms VTM 9.0 (intra) in PSNR and Multiscale Structural Similarity (MS-SSIM).