PART IT. 
LIFE ANNUITIES. 
102. A society consists of 100 persons, 20 of whom are to go out by 
lot every year; each member, at the commencement, is to contribute an 
equal sum to form a fund for the payment of £l at the end of every year 
to each who remains; what is the amount to be contributed by each 
when the interest of money is 4 per cent ? 
At the end of the first year there will be 80 members, each of whom 
is to receive £l; at the end of the second year the number left will be 
60, at the end of the third year 40, at the end of the fourth 20, and at 
the end of the fifth there will be none left; by Art. 33, 
80Xl.04“'=80 x .961538=76.92304 
60 xl.04~ 2 =60x .924556 = 55.47336 
40x 1,04~ 3 =40 X .888990 = 35.55984 
20xl.04“ 4 =20 x .854804=17.09608 
to provide for the payments 
at the end of the 1st year, 
ditto 2nd ditto, 
ditto 3rd ditto, 
ditto 4th ditto, 
their sum ”185.05232 = the total amount to be con 
tributed to form the requisite fund, which, divided by 100 (the number 
of contributors), gives 1.850 = £l 17 0, the sum to be contributed 
by each. 
103. In the Carlisle Rate of Mortality (Table 1), of 10,000 persons 
born, 8461 survive one year, 7779 survive 2 years, 7274 survive 3 years, 
and so on till they all become extinct. If, when the interest of money 
is 3 per cent, it were required to provide at the time of birth ¿£l for each 
of the 10,000 who survive one year, it appears that £8461 would be 
paid amongst them at the end of a year, the present value of which, 
8461 X 1.03“ \ is the sum which will provide for the payment of £l 
to each survivor, which, divided by 10,000, gives 
8461 X 1.03-* 
10000 
the sum to be contributed on behalf of each; if £l is to be provided at 
the time of birth for each child who survives 2 years, 7779 X 1. 03~ 2 is 
the sum to be set apart for the payment of the 7779 who survive that 
7779 x 1.03 -2 
period, and —qqq is the sum to be contributed on behalf 
of each.