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. Ans. 12 years. 7. In how many years will 500 dollars amount to 900, at 5 per cent. compound interest? 8. In what time will 10,000 dollars amount to 16,288, at 5 per cent. compound interest ? 9. At what rate of interest, will 11,106 dollars amount to 20,000 in 15 years Ans. 4 per cent. Ans. 10 years. 10. What principal, at 6 per cent. compound interest, will amount to 3188 dollars in 3 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. 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 281 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 it self, 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, then, as in the preceding article, A=a" XP, and Ex. 1. The population of the United States in 1820 was 9,625,000. Supposing the yearly rate of increase to be 'ith part of the whole, what will be the population in 1830 ? Here P=9,625,000. n=10. a=1tit=1 And log. A=10 xlog. +log. (9,625,000,) 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 jo; wbat 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 išo; 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 eginning of the year ? Here log. a= 10 Therefore, a=1.029; and 267, or 2.9 per cent. is the rate of 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 zo ? Ans. 47 years. 6. If the population of a country, at a given time, be seven millions ; and if the yearly rate of increase be a'th ; what 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 a's ? 8. In what time, will the population of a country advance from four millions to seven millions, if the ratio of increase be to? 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 i's, what will be the number of inhabitants, at the end of 60 years? Here the yearly rate of increase=i's-1's=377. 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 fronı, 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 the ratio of emigration ti and if the population this year be 10 millions, what will it be 20 years hence ? 1 The rate of the natural increase ao- zboi And the population at the end of 20 years, is 12,611,000. 1 37 ) 12. If the ratio of the births be of the deaths, of emigration, in what time will three millions increase to four and a half millions ! 1 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 ; If P= the first term, n=the number of periods ; 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. 2. If the descendants of Noah, beginning with his three sons and their wives, doubled once in 20 years for 300 years, what was their number, at the end of this time? Ans. 196,608. 3. The population of the United States in 1820 being 9,625,000 ; what must it be in the year 2020, supposing it to double once in 25 years ? Ans. 2,464,000,000. 4. Supposing the descendants of the first human pair to double once in 50 years, for 1650 years, to the time of the deluge, what was the population of the world, at that time? EXPONENTIAL EQUATIONS. 62. An EXPONENTIAL equation is one in which the letter expressing the unknown quantity is an exponent. Thus a =b, and x*=bc, are exponential equations. These are most easily solved by logarithms. As the two members of an equation are equal, their logarithms must also be equal. If the logarithm of each side be taken, the equation may then be reduced, by the rules given in algebra. Ex. What is the value of x in the equation 3*=243 ? Taking the logarithms of both sides, log. 3° = log. 243. But the logarithm of a power is equal to the logarithm of the root, multiplied into the index of the power. (Art. 45.) Therefore (log. 3) x x=log. 243; and dividing by log. 3, log. 243_2.38561 –5. So that 35 = 243. log. 3. 0.47712 63. The preceding is an exponential equation of the simplest form. Other cases, after the logarithm of each side is taken, may be solved by Trial and Error, in the same manner as affected equations. (Alg. 503.) For this purpose, make two suppositions of the value of the unknown quantity, and find their errors; then say, |