Any three of these quantities being given, the fourth may be found. Ex. 1. What is the amount of 20 dollars, at 6 per cent. compound interest, for 100 years? Amount of 1 dollar for 1 year 1.06 log. 0.0253059 2. What is the amount of 1 cent, at 6 per cent. compound interest, in 500 years? Amount of 1 dollar for 1 year 1.06 log. 0.0253059 More exact answers may be obtained, by using logarithms of a greater number of decimal places. 3. What is the amount of 1000 dollars, at 6 per cent. compound interest, for 10 years? Ans. 1790.80. 4. What principal, at 4 per cent. interest, will amount to 1643 dollars in 21 years? Ans. 721. 5. What principal, at 6 per cent., will amount to 202 dollars in 4 years? ។ Ans. 160. 6. At what rate of interest, will 400 dollars amount to 569, in 9 years? Ans. 4 per cent. 7. In how many years will 500 dollars amount to 900, at 5 per cent. compound interest? Ans. 12 years. 8. In what time will 10,000 dollars amount to 16,288, at per cent. compound interest? Ans. 10 years. 9. At what rate of interest, will 11,106 dollars amount to 20.000 in 15 years? Ans. 4 per cent. 10. What principal, at 6 per cent. compound interest, will amount to 3188 dollars in 8 years? Ans. $2000. 11. What will be the amount of 1200 dollars, at 6 per cent. compound interest, in 10 years, if the interest is converted into principal every half year? Ans. 2167.3 dollars. 12 12. In what time will a sum of money double, at 6 per cent. compound interest? Ans. 11.9 years. 13. What is the amount of 5000 dollars, at 6 per cent. compound interest, for 28 years? Ans. 25.942 dollars. INCREASE OF POPULATION. 61. b. The natural increase of population in a country, may be calculated in the same manner as compound interest; on the supposition, that the yearly rate of increase is regularly proportioned to the actual number of inhabitants. From the population at the beginning of the year, the rate of increase being given, may be computed the whole increase during the year. This, added to the number at the beginning, will give the amount, on which the increase of the second year is to be calculated, in the same manner as the first year's interest on a sum of money, added to the sum itself, gives the amount on which the interest for the second year is to be calculated. = If P the population at the beginning of the year, a 1+the fraction which expresses the rate of increase, nany number of years; and A= the amount of the population at the end of n years; then, as in the preceding article, Ex. 1. The population of the United States in 1820 was 9,625,000. Supposing the yearly rate of increase to be th part of the whole, what will be the population in 1830? Here P 9,625,000. n=10. a=1+3=3 And log. A 10xlog. +log. (9,625,000,) 4 2. If the number of inhabitants in a country be five millions, at the beginning of a century; and if the yearly rate of increase be; what will be the number, at the end of the century? Ans. 132,730,000. 3. If the population of a country, at the end of a century, is found to be 45,860,000; and if the yearly rate of increase has been ; what was the population, at the commencement of the century? Ans. 20 millions. 4. The population of the United States in 1810 was 7,240,000; in 1820, 9,625,000. What was the annual rate of increase between these two periods, supposing the increase each year to be proportioned to the population at the beginning of the year? Here log. a= log. 9,625,000-log. 7,240,000 10 Therefore, a=1.029; and, or 2.9 per cent. is the rate uf increase. 5. In how many years, will the population of a country advance from two millions to five millions; supposing the yearly rate of increase to be? Ans. 47 years. th; what 6. If the population of a country, at a given time, be seven millions; and if the yearly rate of increase be will be the population at the end of 35 years? 7. The population of the United States in 1800 was 5,306,000. What was it in 1780, supposing the yearly rate of increase to be? 8. In what time will the population of a country advance from four millions to seven millions, if the ratio of increase be T? 9. What must be the rate of increase, that the population of a place may change from nine thousand to fifteen thousand, in 12 years? If the population of a country is not affected by immigration or emigration, the rate of increase will be equal to the difference between the ratio of the births, and the ratio of the deaths, when compared with the whole population. Ex. 10. If the population of a country, at any given time, be ten millions; and the ratio of the annual number of births to the whole population be, and the ratio of deaths, what will be the number of inhabitants, at the end of 60 years? 360° Here the yearly rate of increase= The rate of increase or decrease from immigration or emigration, will be equal to the difference between the ratio of immigration and the ratio of emigration; and if this differ ence be added to, or subtracted from, the difference between the ratio of the births and that of the deaths, the whole rate of increase will be obtained. Ex. 11. If in a country, the ratio of births be the ratio of deaths 301 1 the ratio of immigration, and if the population this year be 10 millions, what will it And the population at the end of 20 years, is 12,611,000. in what time will three millions increase to four and a half millions? If the period in which the population will double be given; the numbers for several successive periods, will evidently be in a geometrical progression, of which the ratio is 2; and as the number of periods will be one less than the number of terms; Ex. 1. If the descendants of a single pair double once in 25 years, what will be their number, at the end of one thousand years? The number of periods here is 40. |