What time periods did the penny contain the same percentage of copper as nickel?

The US cent never contained equal amounts of copper and nickel. In fact US cents only contained nickel from 1856 to mid-1864, in the ratio 12% nickel to 88% copper.

That alloy proved difficult to strike without unacceptably high die breakage; as a result the cent's composition was changed to a bronze alloy of 95% copper and 5% tin and/or zinc. Bronze was used until 1982 when the current copper-plated zinc composition was adopted. The only exception occurred in 1943 when wartime copper shortages forced the Mint to strike cents in steel.