EXAMPLES: (1) Find the compound interest of $525 for 1 year 6 months, at 5%, payable quarterly. Here Pr and n are given and A required. Performing the operation indicated by means of logarithms, we have: A = $525 (1+ 5(1+.05) 0 Hence, compound interest = $40.62. = $565.62. (2) Find the present value of $1000, due in 5 years, at 10% compound interest. Annuities. An annuity is a fixed sum of money to be paid under stated conditions at equal intervals of time. It may be: Certain, i.e. payable independent of any condition. Life, i.e. payable during the life of some certain individual. Deferred, i.e. payable after the lapse of a number of years. Perpetual, i.e. payable forever. 434. III. To find the present worth of an annuity for a number of years, allowing compound interest. 435. Let A = Annuity in dollars. R: = Amount of $1 for 1 year. P = Present worth of annuity. By Art. (432), annual payments, APR at end of 1st year. (1) Find the present worth of an annuity of $500 to continue for 15 years, with compound interest at 4 per cent. Here, A = 500, n = 15, R = 1.04, R−1 = .04, which applies when the present worth and the amount of one dollar for one year are given. If in formula (6) n is increased indefinitely, the limiting value of the second member becomes That is, the present worth of a perpetual annuity is equal to the amount divided by the rate. From formula (6) we again have which applies when the present worth of an annuity is required, which begins after p years and continues for n years thereafter. From formulas (10) and (8) we obtain which applies when the present worth of an annuity is required, which begins after p years and continues in perpetuity. EXERCISE 126. 1. Find the interest of $350 for 10 years at 7 per cent, simple interest. 2. Find the compound interest of $1000 for 6 years at 5 per cent. 3. What will be the amount of $1200 at 8 per cent for 10 years, compound interest? 4. What sum of money will amount to $1050 in 4 years at 3 per cent, compound interest? 5. What is the present worth of a note of $1580, due in 12 years at 5 per cent, interest compounded semiannually? 6. At what rate per cent yearly will $1850 amount to $2275 in 4 years and 6 months, interest compounded quarterly? 7. In how many years will $200 double itself at 6 per cent, annual compound interest? 8. In how many years will a sum of money quadruple itself at 10 per cent, semiannual compound interest? 9. What is the present worth of an annuity of $400, to continue 20 years at 7 per cent, compound interest? 10. What is the present worth of an annuity of $1250, to continue at 4 per cent for 20 years, interest compounded annually? 11. What is the present worth of a perpetual annuity of $150, to begin after 8 years, annual compound interest at 5 per cent? 12. What annuity can be purchased for $1284.70, if it is to run for 14 years at 6 per cent, annual compound interest? CHAPTER XXXVI. CONTINUED FRACTIONS. 437. A Continued Fraction is an expression of the following form: 438. Any proper fraction in its lowest terms may be converted into a continued fraction by the method of finding the G. C. D. 439. If the number of the denominators of the fraction is finite, it is called a Terminating continued fraction. 440. If the number of denominators of the fraction is indefinitely great, it is called an Infinite continued fraction. Convergents. In the continued fraction 1 a, is called the first convergent a1+ is called the second 1 1 convergent, a1 + a2 + so on. Let is called the third convergent, and be any fraction in its lowest terms. |