The half life of radium is given as 1600 years. This means that a sample of 1 kg or 1000 g radium will be reduced to 500 gm in 1600 years. After another 1600 year years, it will be reduced further by another 50% and we will be left with 250 g of radium and so on. In other words,
`(1/2)^n` = fraction of original sample left
where, n is the number of half lives. Thus, in 1 half life, 1/2 of original sample is left. In 2 half lives, 1/4 or 25% of original sample is left and so on.
In this case, we have to find the time after which only 1 g of sample is left. In other words, we have to find the value of n for which only 0.001 part (= 1 g/1000 g) of radium is left.
Hence, (1/2)^n = 0.001
solving this (hint:take logarithm of both sides and then solve), we get, n = 9.97 half lives. = 9.97 x 1600 years = 15,952 years.
Thus, after a period of 15,952 years, 1 kg of radium will be reduced to 1 g of radium.
Hope this helps.