Nampak Cartons produces cartons for the food and beverages sectors from sites including Leeds and Gillingham in the UK, where it employs some 600 staff.
Nampak Healthcare manufactures products for the pharmaceutical industry from plants including Bradford, Bromborough, Liverpool, Newmarket and Portsmouth, and it has 464 staff in the UK.
A Nampak spokeswoman said: "It is envisaged that the management teams and staff of Nampak Cartons and Healthcare will move with the businesses which will be re-branded as part of the change in ownership."
In a statement, Nampak said the reason behind the change of ownership is that the African company has "recently embarked on a strategy of growing and investing in what it believes to be its core business for the future".
It said: "As a result of this strategic review, there have already been some divestitures in South Africa and the sale of these two businesses in Europe is a further progression of that strategy."
The purchase consideration is based on an enterprise value of £80m, and after deducting pension fund liabilities, quasi debt items and other miscellaneous transaction adjustments, the net proceeds are estimated to be £65m.
In PrintWeek's Top 500 2010, Nampak Holdings (UK) was recorded as having a turnover of £346m, compared with £348m at the same time a year earlier.
Platinum Equity is an investment company that was founded in 1995 by Tom Gores and it has so far completed over 100 acquisitions.
It has offices in Boston, New York and London, and this is its first acquisition in packaging.