log. n= 14.334 nearly Ans. 1.156356 5. What will be the compound interest of $947 for 4 years and 3 months at 54 per cent. ? 6. What will $157.63 amount to in 17 years at 4 per cent.. 7. A note was given the 15th of March 1804, for $58.46, at the rate of 6 per cent. compound interest; and it was paid the 19th of Oct. 1823. To how much had it amounted ? 8. A note was given the 13th of Nov. 1807, for $456.33, and was paid the 23d of Sept. 1819. The sum paid was $894.40. What per cent. was allowed at compound interest : 9. In what time will the principal p be doubled, or become 2 p, at 6 per cent. compound interest? In what time will it be tripled? Note. In order to solve the above question, put 2 p in the place of A for the first, 3 p for the second, and find the value of n. The principles of compound interest will apply to the following questions concerning the increase of population. 10. The nuinber of the inhabitants of the United States in A. D. 1790 was 3,929,000, and in 1800, 5,306,000. What rate per cent. for the whole time was the increase ? What per cert per year? 11. Suppose the rate of increase to remain the same for the next 10 years, what would be the number of inhabitants in 1810? 12. At the same rate, in what time would the number of inhabitants be doubled after 1800 ? 13. The number of inhabitants in 1810 by the census was 7,240,000. What was the annual rate of increase ? 14. At the above rate, what would be the number in 1820 ? 15. At the above rate, in what time would the number in 1810 be doubled ? 16. The number of inhabitants by the census of 1820, was 9,638,000. What was the annual rate of increase from 1810 to 1820 ? 17. At the same rate, what is the number in 1825 ? 19. At the same rate, in what time will the number in 1820 be doubled ? 20. In what time will the number in 1820 be tripled ? 21. When will the number of inhabitants, by the rate of the last census, be 50,000,000 ? LIII. 1. Suppose a man puts $10 a year into the savings bank for 15 years, and that the rate of interest which the bank is able to divide annually is 5 per cent. How much money will he have in the bank at the end of the 15th year? Suppose a = the sum put in annually, r = the rate of interest, the time, According to the above rule of compound interest, the sum a at first deposited will amount to a (r + 1); that deposited the second year will amount to a (r + 1).-.; that deposited the third year will amount to a (r + 1)?; that deposited the last year will amount to a (r + 1)'. Hence we have A=a (r + 1) + a(r + 1)^ ta(r+1)-2.... a (r+1) .... a[(r + 1) + (r + 1)e- + (r + 1)e-?.... (r + 1)] But (r + 1), (r + 1)-, &c. is a geometrical progression, whose largest term is (r + 1)', the smallest r + 1, and the ratio r +1. The sum of this progression, Art. XLVII. is (n+1) [(r +1)– 1] The same result may be obtained by another course of reasoning The amount of the sum a for one year is a + ar. Adding a to this, it becomes 2 a + a r. The amount of this at the end of another year is 2 atar + 2 ar tam, or 2 a + 3 ar + ar?. Adding a to this it becomes 3 a + 3 a r t a r?. The amount of this for 1 year is = 3 a + 6 ar + 4 a pi + a mi, = a (3 + 6 r + 4 m2 + ry). This is the amount at the end of the third year before the addition is made to the capital. The law is now sufficiently manifest. With a little alteration, the quantity 3+ 6 r + 4 gi? + på may be rendered the 4th power of 1 +r. The three last coefficients are already right. If we add 1 to the quantity it becomes 4 + 6 + 4r + r. Multiply this by r and it becomes 4r + 6 pl + 4 + r. Add 1 again and it becomes 1+4r + 6 p + 4 pued + pt. This is now the 4th power of 1 + 1, and it may be written (1 + r)* Subtract the 1 which was added last, and it becomes (1 + r) — 1. Divide this by r, because it was multiplied by r, and it be. (1 + r)* — 1 comes r Subtract 1 again, because I was added previous to multiplying by r; and it becomes (1+r)*—1 (1+r)^(1+r) – (1+r) [(1+r)—1] 1 Substitute t in place of the exponent 3, and multiply by a, and it becomes a (1+r) [(1 + r)e—1] = A. r which is the same as before. The particular question given above may now be solved by logarithms, using this formula. log. (1 + r) = 1.05 0.021189 Multiply by t = 15 15 2. A man deposited annually $50 in a bank from the time his son was born, until he was 20 years of age ; and it was taken out, together with compound interest on each deposit at 3 per cent., when his son was 21 years of age, and given to him. How much did the son receive ? 3. How much did the bankers gain by receiving the money, supposing they were able to employ it all the time at 6 per cent. compound interest? 4. A man has a son 7 years old, and he wishes to give him $2000 when he is 21 years old ; how much must he deposit annually at 4 per cent. compound interest, to be able to do it? 5. If a man deposits in a bank annually $35, in how long a time will it amount to $500 at 6 per ceat. compound interest ? 6. The first slaves were brought into the American Colonies in the year 1685. Suppose the first number to have been 50, and that 50 had been brought each year for 100 years, and the rate of increase 3 per cent. How many would there have been in the country at the end of the hundred years ? LIV. Annuities. 1. A man died leating a legacy to a friend in the following manner ; a sum of money was to be put at interest, such that, the person drawing 10 dollars a year, at the end of 15 years the principal and interest should both be exhausted. What sum must be put at interest at 6 per cent. to fulfil the above condition? Let the learner generalize this example and form a rule; and then solve the following examples by it. 2. A man wishes to purchase an annuity which shall afford him $300 a year so long as he shall live. It is considered probable that he will live 30 years. What sum must he deposit in the annuity office to produce this sum, supposing he can be allowed 3 per cent. interest ? N. B. The principal and interest niust be exhausted at the end of 30 years |