how to resize image so that it becomes icon

I have an image which is of dimensions 490*307. This image is to be used as an icon of size 24px*15px for the website. How to resize the image so that it becomes of the smaller size without losing the quality of the image? I want the solution for many images(provided by designers) which are too large in size but the size is to be reduces so that it can be used as logo or icon on the html website.

I have used many online free tools. But none of them result in required quality. All the images are blurred enough to e used as the icon. I don't have any paid software such as Photoshop etc.