Note. When the payments are to be made half yearly or quarterly the present worth so found must be multiplied by the proper number in Table V. Questions. What are annuities at compound interest? When the annuity, time and rate of interest are given, by what rule do you find the amount? What is to be noticed when the payments are half yearly or quarterly? When the annuity, time and rate are given to find the present worth, how do you proceed? What is to be noticed when the payments are half yearly or quarterly? ANNUITIES IN REVERSION. Sums of money which are payable yearly for a limited period, but which do not commence till after the expiration of a given period are called annuities in reversion. The annuity, time of reversion, time of continuance and rate given to find the present worth of the reversion, Rule. Take two numbers under the given rate in Table IV., that opposite the sum of the two given times; and the number opposite the time when the annuity is to commence, or time of reversion, and multiply their difference by the annuity for the present worth. Note. When the payments are to be half yearly or quarterly, use Table V, as before. Questions. What are annuities in reversion? When the annuity, time of reversion, time of continuance, and rate are given to find the present worth, by what rule do you work? What is to be noticed when the payments are half yearly or quarterly? PERPETUITIES AT COMPOUND INTEREST. Annuities which continue for ever are called perpetuities. The annuity and rate given to find the present worth. Rule. Divide the annuity by the ratio less 1 for the present worth. Note.-Table V. must be used as in temporary annuities when the payments are half yearly or quarterly. Questions. What name is given to annuities which continue for ever? By what rule do you proceed when the annuity and rate are given to find the present worth? What is to be noted when the payments are half yearly or quarterly? COMPOUND INTEREST BY DECIMALS, Examples. 1. What is the interest and amount of 4001. for 3 years at 4 per cent ? 1.04 X 1.04 X 1.04 1.124864 400 449 945600 amount. 49.9456 interest. 2. What is the amount and interest of 750l. at 5 per cent per annum, for 4 years 6 months? Ans. Amount 934/. 23. 10d., interest 1847, 28. 10d. Case 2. 1. What principal put to interest will amount to 6957. 138. 9d. in 5 years at 5 per cent,? Ans. 545l, 18. 11d. 2. What principal must be put to interest to amount to 260l. 58. 3d. ot 6 per cent. per annum for 3 years? Ans. 2187. 108, 5d. ANNUITIES AT COMPOUND INTEREST. Case 1. 1. What is the amount of an annuity of 180 dollars for 9 years at 5 per cent.? 11.026564 180 882125120 11026564 $1984.781520 Ans. 2. What will annuity of $200, amount to in 5 years to be paid by half yearly payments, at 6 per cent. per annum? Ans. $1144.08 2m.† Case 2. 1. What is the present worth of 501, per annum for 6 years at 4 per cent.? 5.24214 50 1.262.10700 2. What is the present worth of 70 dollars a year for 5 years payable yearly, half yearly and quarterly, at 6 per cent. per annum? Ans. Yearly $294.86 5m. + Half yearly $299.22 3m.+ Quarterly $301.42 8m.+ ANNUITIES IN REVERSION. 1. The reversion of a freehold estate ef 60%. per annum, for 4 years, to commence 2 years hence, what is the present worth allowing 4 per cent. for present payment? 5.24214 1.88609 3.35605 60 L. 201.36300 Ans. 2. What is the present worth of a reversion of a lease for $120 per annum, to continue 9 years, but not to com. mence till the end of 4 years, at 4 per cent. to the purchaser? Ans. $762.69 1m.+ PERPETUITIES AT COMPOUND INTEREST. 1. What is the present worth of an annuity of 1607. per annum to continue for ever, allowing 5 per cent. to the purchaser? 1.05-1.05)150.00 $3000 Ans. 2. What is an estate of 260 dollars per annum, to continue for ever, worth in present money, allowing 6 per cent. to the purchaser ? Ans. $4333.33 3m.+ COMBINATION. Combination is used to show how many different ways a less number of things can be combined out of a greater as out of the figures 1, 2, 3, the three combinations 12, 13, and 23, may be formed. Rule. 1. Take a series proceeding from and increasing by a unit up to the number to be combined. 2. Take another series of as many places decreasing by unity from the number out of which the combinations are to be made. 3. Multiply the first continually for a divisor and the latter for a dividend, the quotient will be the an swer. By what rule do you work questions in Combination? Examples. 1. How many combinations of 3 persons in 6? 6X5X4 1 X2 X3 -20 Ans. 2. How many combinations of 10 figures may be made out of 20? Ans. 18302. PERMUTATION. Permutation us used to find how many different ways a given number of things may be varied in succession as 123, 132, 213, 231, 312, 321 are six different permutations of three figures. Rule, Multiply all the number continually in succession, from one to the given number inclusive, the product will be the number of variations. Questions. What is Permutation? What is the rule for finding the number of variations in any given number? Examples. 1. In how many different positions can 7 men place themselves round a table? 1 X2 X3 X4 X5 X6 X 7=5040 Ans. 2. In what time will a person make all the changes that the 12 first letters of the alphabet admit of; allowing 15 seconds for each change? Ans. 8870 hours, 24 min. or, 1 year 4days 14 hours 24 min. |