The white minority was a bit larger in the early era, and even during apartheid itself; certainly the spatial aspects of apartheid were designed to "make" areas whiter on the surface. But how did the nonwhite areas get cordoned and contained so? The short answer is that apartheid didn't just happen one day, but was the outgrowth of a variety of much earlier developments. Those developments extended from the Colony of the Cape of Good Hope (VOC/Dutch/Batavian and British) which had a much larger percentage of Europeans present, and by the time they got to lock horns with the major existing kingdoms in the 19th century, they were militarily fairly powerful--what ensued was the creation of a colonial system and colonial relations, taken arguably to its highest form (per Mamdani's Citizen and Subject, though people disagree with some of his ideas).
The development of the colony to its greatest extent, and the development of various settler republics (and metis republics like the Griqua), happened in the context of the selective application of force but also very importantly the incorporation of African allies into systems of rule. Various kingdoms and smaller groups of Bantu-speaking people found advantage at different times in working with settlers, be they Boer or English, and were tolerated or even welcomed in turn at least for a while. Only after the 1880s was it quite clear that the various groups of settlers formed something unified relative to non-whites; before that time, it was possible to get (for example) baSotho to aid Boers against the British, as happened in the Orange River Sovereignty in the early 1850s. Certainly, during the South African War, a lot of Africans hoped that British victory would bring liberalization, when in fact it cut them out, and ultimately led to the creation of the first proper national [black] political organizations after the Bloemfontein SA Native Convention in 1909 [not 1908; corrected]. (See Andre Odendaal's The Founders on the early process--it's really fascinating, although he does spend most of his time on the Eastern Cape.)
At that point, however, the balance of force rested (as elsewhere in Africa) almost entirely on the side of the Europeans. For a brief time, they (British and non-British) had a monopoly on weapons that permitted the investiture of defensive positions and the assault of strongholds with impunity, and they used it to break any overt resistance as in the Transkei around 1880, the northern Transvaal during the 1890s, and KwaZulu in 1906. In the wake of those selective attacks and incorporations, initially the practice of the various colonies (and after May 1910, the Union of South Africa) was to weaken chiefs and kings and place them in "reserve" areas that hopefully would dissipate as "tribalism" withered. This changed when their value to segregationism as indirect rulers became clear, and African communities in turn saw the opportunity to mediate the effects of acquisitive settler activity without bloodshed through some form of local self-governance. Conveniently, providing that to non-whites also permitted them to be excised fully from the governance of the country as a whole, though the last black voters would only vanish from the rolls in 1936.
So a model of invasive but nominally indirect rule followed British segregationist models tied to a sense of South African, mostly Afrikaner, nationalism by World War II (per Saul Dubow's germinal Racial Segregation and the Origins of Apartheid in South Africa 1919-1936 and Paul Rich's study of E. T. Stubbs in African Affairs in 1980). The trouble was that during that entire period the SA Native National Congress (after 1923 renamed the ANC, as today) was having enormous trouble casting itself as a party of mass action and unity, instead of one that represented only a narrow educated elite. They had some success with trade unions and building bridges, but only with the rise of the Youth League and then apartheid itself did a true mass party for the black population come into its own. (Peter Limb's The ANC's Early Years is the best text on this era, hands down.) So unlike the whites, nonwhites didn't have the same kind of political unity, and had to compete with newly re-invented and re-elevated traditional leadership (based on the divisive colonial model of tribalism) much of the time.
By the time apartheid's technical machinery came into being with the flurry of acts championed by Hendrik Verwoerd from 1950 to 1966 (PM from 1958-1966), then, the SA government had already devolved much of its direct political action to local "traditional" leaders and black-run self-government structures in legally defined areas. These became the hated Bantustans and townships under the Nasionale Party apartheid regime. Again, the government maintained as far as possible a monopoly on the means of violence, as well as a healthy distance from the effects of their actions. But after World War II, action to fight colonial power began almost immediately--first in words, then in protests, and finally in the armed struggle's several major phases once it was clear there was no true negotiating for majority rule possible--but it worked in effect the same way that any colonial regime functioned, just here the colonialism was internal. A great many observers noticed this from day one, and even Verwoerd's own hand-picked commissions (notably the Tomlinson Commission) repeatedly pointed out this fact as an intractable problem with apartheid's claims of separate development. (Dubow's new general history Apartheid 1948-1994 and Welsh's The Rise and Fall of Apartheid are both useful to read for this process overall, and the distance between theory, reality, and practice. Dubow's better I think on the early phases and the theory, while Welsh is excellent at political and social questions especially in the emergent conditions of late apartheid.)
It's also worth pointing out that the colonial states (and their apartheid derivative) were never able to actually exert the control they claimed. They followed the legal forms, and could impose their will in particular spots, but the edges of their power were really very visible in some areas. For example, parts of Venda were never really under South African state control--the state could intervene if they really wanted to, but they never stayed, and in fact overlooked a lot of "squatting" and "illegal" gatherings in favor of monitoring the northern border later on. Verwoerd also wanted to wipe out the growing townships surrounding major cities and de-nationalize everyone, but over time, he [and later Nats] had to accept their continued existence in the face of opposition from industry and other white employers, but also the incredible cost and bad press that would be involved in destroying them, which was bad enough where it did happen [see "forced removals" and "black/white spots"]. The Kruger Park, to give yet another example, was only manned with great cost and difficulty during the 1970s and 1980s, yet enemies of the apartheid state could still come across the border easily. As soon as serious resistance organized, and especially after 1976 when that resistance forced the state to show its fangs in a particularly ugly way in Soweto, the control of the periphery by the apartheid state began to fall apart even more rapidly. It was always an invasive and onerous regime in both "petty" (daily social/economic) and "grand" (political and national) varieties, but it never worked consistently the way its proponents suggested it could or would.
[edit: italicization fixed; a couple of sentences; one more title; accidentally a few words; added a bit of text to Welsh/Dubow and corrected SANC year]
Khosikulu provides a great answer and I just wanted to add a few a thoughts.
It seems that your question is one about numerical logistics. You're thinking that for example, if there are 10 people in a room, technically 1 person shouldn't be able to control 9.
But what we've seen throughout history is that effective means of control of populations has little to do with number of people in charge or their proportion relative to the 'rest'. Rather its the structures, monopolization of force and mentality of the divided peoples.
Whether in Sunni Iraq (which is Shia majority) or in French Algeria (majority muslim, ruled by France before independence) a lot of groups have been able to obtain and keep power by utilizing those three factors mentioned previously.
The British understood this better than anyone and practiced it time and time again
For example:
India - The British were able to effectively control all spheres (political, military, economic) in India with extremely small numbers. The 1931 census shows that ~160,000 British lived in India. At the same time, there were approximately 278,000,000 Indians!
Jamaica - By the beginning of the 19th century, due to the country's dependence on slavery for its sugar industry, there approximately 20 blacks for every 1 white.
How was this possible? Well in both places the British had the absolute monopoly of force. They had superior organization and an effective law & punishment systems (which favored the status quo) and a very astute understanding of local mentality that allowed rule to persist for as long as it did.
Hi there, there's a great deal to be said on this topic. While you wait for an expert to chime in, you may be interested in some of these older threads on apartheid in South Africa and elsewhere:
http://www.reddit.com/r/AskHistorians/comments/1qozjx/ama_history_of_southern_africa/
http://www.reddit.com/r/AskHistorians/comments/1iwtme/apartheid_vs_american_segregation/
If you find something interesting in them, please ask follow-up questions here and mention usernames so that those users will be alerted to your question.