Provide one important similarity and one important difference between the way the British dominated India, and the way they dominated China.

The most important difference between the British domination of India and China is that the British Empire fully and directly colonized India under a Mercantile system known as the British Raj, whereas the British never took full or direct control of the Chinese government or economy. 

In India, the British instituted an enormously powerful and far reaching colonial regime, run by English governors and soldiers, who reported directly back to London. The British built up a massive and complex infrastructure, and even instituted the rigid Caste System, which is a perfect example of a "divide and conquer strategy."

In China, on the other hand, instead of fully overtaking the levers of power, the British undermined and coopted the weakened Qing Dynasty, using force and the threat of force to gain favorable trading terms with Chinese Merchants. Although the British gained full control over Hong Kong in the aftermath of the Opium Wars, they never dominated the mainland, but they didn't have to.

Instead, the British got what they wanted through a less blatant use of "divide of conquer strategy," which required a smaller colonial footprint than in India. The divide and conquer strategy used by the British in China had less to do with turning various segments of the Chinese populace against one another (as in India) and had more to do with making strategic alliances with other colonial powers interested in China, most notably Japan, Russia and the United States. By agreeing to an "Open Door Policy" of trade with these competing countries, Britain insured that their businesses could maintain access to Chinese markets. 

Although the methods of British subjugation in China and India different in their specifics, in both cases the British successfully pitted their adversaries against each other, fomenting internal strife that made the populations easier to control.