British Citizen travelling to Beijing, Hong Kong & Singapore

I am travelling with my Dad and Brother from London to Beijing for 4 nights then to Hong Kong for 4 nights then Singapore for 4 nights and back to London from Singapore. We have paid £151 each for the L Chinese Visa and was wondering if we needed to as I have heard about the free transit entry. Please can you explain how this works and how many days you are allowed within the free transit entry?