2. Find the amount of an annuity of $50, for 7 yrs. at 5 per ct. Ans. $402.50. 3. What is the amount of an annuity of $250 for 7 yrs. at 6 per ct. payable half yearly? Ans. $2,091.25. NOTE. As the annuity is payable half yearly, it amounts to the same sum it would, if the years were twice as many, and the rate half as great. 4. If a rent of $200 remain unpaid 8 years, what will it amount to? Ans. $1,936. 5. A man sold a pension of $100 to continue 5 years, allowing the buyer 6 per ct. What did he receive? We may consider the annuity as five separate debts, due at the end of 1, 2, or 3 years, &c. Discounting at 6 per ct. on each, (§ LXXXII) we have 1st. year, present worth, $94.3396; 2d. year, $89.2857; 3d. $84.7457; 4th. $80.6451; 5th. $76.9230. The amount of these, 425.9391, is the amount of the annuity, required. Hence, to find the present worth of an annuity, for a given time, OF THE PRESENT WORTHS THUS FOUND WILL BE THE ANSWER. 6. Find the present worth of a pension of $500 for 4 years at 5 per cent. Ans. $1,782.185+ 7. Find the present worth of a salary of $200, to continue 3 yrs. at 4 per ct. Ans. $556.063. NOTE. The estimation of the present worths of Annuities at Simple Interest, may easily be shown to be unreasonable and unjust. For the price of an annuity of $100 found in this way, for 40 years, at 6 per ct., will amount to a sum, which, put at interest, would draw a greater annuity than the given one; and one which would last forever, instead of forty years. It is therefore most equitable to allow Compound Interest. ANNUITIES AT COMPOUND INTEREST. 1. An annuity of $100 was left 4 years unpaid. What was then due on it at 6 per ct. compound interest? In finding an amount at compound interest, we multiply continually by the rate per ct.+1, (§ LXXXVIII) till the number of multiplications is equal to the number of years. Now, in calculating an annuity, we have for the last year, the annuity without interest; for the next preceding, the amount of the annuity for 1 yr.; for the next preceding still, the amount for 2 yrs., and so on. Thus, at compound interest, an annuity in arrears forms a Geometrical Progression, whose ratio is the rate per ct.+1, and whose terms equal the years in number. In the above example then, 100×1.06=119.101. 119.101× 1.06—100÷.06=$437.45 Ans. Hence, to find the amount of an annuity in arrears at compound interest, 3 FIND THE SUM OF A GEOMETRICAL SERIES, WHOSE FIRST TERM IS THE ANNUITY, AND WHOSE RATIO, THE RATE PER CENT.+1, FOR AS MANY TERMS AS THERE ARE YEARS. THIS WILL BE THE AMOUNT REQUIRED. 2. Find the amount of $150 annuity, for 4 years, at 10 per cent. A. $696.15 3. Find the amount of an annuity of $40, for 5 years, at 5 per cent. A. $221.02525. 4. Find the amount $50 annuity, for 7 years, at 4 per cent. A. $394.915 For convenience, has been calculated the following TABLE OF MULTIPLIERS, FOR FINDING THE AMOUNt of an annUITY, FOR ANY NUMBER OF YEARS, Yrs.16 per cent. Yrs.16 per cent. Yrs.16 per cent. Yrs. 6 per cent. 5 5. Find the amount of an annuity of $150, for 3 years, at 6 per A. $477.54. cent. 6. Find the amount of $500 annuity, for 24 years, at @ per cent. A. $25,407.75. 7. Find the amount of $100 annuity, for 49 years, at 6 per cent. A. $26,172.08. NOTE. If the time given be greater than 40 yrs., calculate for 40, and consider this amount a debt at compound interest, for the remaining time, which calcu late accordingly. Then calculate the amount for the remaining time, as though the annuity commenced again, and add this sum to the last. 8. A pension of $100 for 4 years, was sold, the buyer being al. lowed 6 per ct. compound interest, for his money. What did the seller receive ? It is evident that the present worth is a sum, which, at compound interest would, in 4 years, produce the amount of the given annuity, for the same time. This amount is $437.46. To find an amount at compound interest, we multiply a sum by the rate per cent. +1, as many times successively as there are years, or, in other words, - we multiply by that power of the rate per cent.+1, whose index is the number of years. To find a present worth, we must, manifestly, reverse this process, and divide by the same power. Then $437.46+1.26247 (=1.064) $346.511 Ans. Hence, to find the present worth of an annuity, FIND THE AMOUNT IN ARREARS FOR THE GIVEN TIME, AND DIVIDE IT BY THAT POWER OF THE RATE PER CENT.+1, WHOSE INDEX IS THE NOTE. This power may be found in the table of multipliers for compound interest. (SLXXXVIII.) 9. Find the present worth of a $40 annuity, to continue 5 yrs. at 5 per cent. A. $173.173. 10. Find the present worth of $100 annuity, for 20 yrs. at 5 per A. $1,246.22. cent. For convenience, has been calculated the following TABLE OF MULTIPLIERS, FOR FINDING THE PRESENT WORTH OF AN ANNUITY FOR ANY NUMBER OF YEARS, FROM 1 TO 20, AT 5 AND 6 PER CENT. 9.89864 9.29498 4 3.54595 3.46510|| 14 5 4.32948 4.21236 15 10.37966 9.71225 6 5.07569 4.91732| 16 10.83777 10.10589 7 5.78637 5.58238 17 11.27407|10.47726| 8 6.46321 6.20979 18 11.68958 10.82760 9 7.10782 6.80169| 19 |12.08532 11.15811 10 7.72173 7.36008|| 20 |12.46221|11.46992 11. Find the present worth of an annuity of $21.54, for 7 yrs. at 6 per cent. A. $120.244+ 12. Find the present worth of an annuity of $100, to continue 12 years, at 6 per cent. A. $838.384. 13. Find the present worth of an annuity of $936, for 20 yrs. at 5 per cent. A. $11,664.629 As any annuity multiplied by one of the numbers in the last table will give the present worth of that annuity, so it is evident that, any present worth, divided by the same number, will give the annuity itself. Hence, if I wish to discover of what annuity any given sum is the present worth, that is, what annuity any given sum will buy, I have only to use the above table, as a table of divisors, instead of multipliers. 14. What annuity, to continue 5 years, will $432,948 purchase, when money is worth 5 per cent. ? A. $100,000. 15. What annuity, to continue 19 years, will $6,694.866 pur-" chase, when money will bring 6 per cent.? A. $600. When an annuity is to commence at some future time, it is said to be in reversion. An annuity in reversion, is evidently not worth as much as one of the same amount, which commences immediately. For if the present worth of the annuity be calculated as usual, it will be what the annuity is worth at the time it commences; and as that time is still future, we must discount for the intervening space, in order to obtain the true present worth. Hence, to find the present worth of an annuity in reversion, FIND THE PRESENT WORTH AS USUAL, AND DISCOUNT UPON IT FOR THE TIME OF REVERSION. NOTE. Of course, the discount should be made as directed in § LXXXVIII, page 230, article COMPOUND INTEREST. Or, the present worth may be calculated as though the annuity were to commence immediately, and to continue to the end of the time of the given annuity: if, from this sum, the present worth for the time of reversion be subtracted, the remainder will be the present worth required. 16. If an annuity of $100 be 14 years in reversion, to continue 20 years afterwards, what is its present worth, discounting at 5 per cent. ? A. $629.426. 17. What is the present worth, at 6 per cent. of an annuity of $120, to continue forever? A. $2,000. NOTE. The answer is evidently a sum whose annual interest is $120. 18. Which is preferable, an annuity of $100 for 15 years, to com. mence immediately, or the reversion of the same annuity, forever, after the 15 years have expired? also, what is the difference? A. The term of 15 yrs. is better than the reversion forever after, by $75.928+ NOTE. If the time extend beyond the limits of the table, calculate as far as the table will allow, and consider the rest as an annuity in reversion. 19. Find the present worth of an annuity of $400, to continue 34 yrs. at 6 per cent. A. $6,477.16. 20. Find the present worth of a $70 annuity, to continue 59 yrs. at 5 per cent. A. $1,321.3021. NOTE. To give a complete developement of the subject of annuities, is not the province of arithmetic. Contingent annuities, or those whose continuance depends on uncertainties, as the duration of the life, or lives, of one, or of several persons, involve the doctrines of CHANCES, and are, in many cases, complex and tedious in calculation. PERMUTATION AND COMBINATION. § CV. The two letters, A B, may be written A B, or BA. Any two things, therefore, have two orders of succession, or relative positions, in which they may be placed, in a single line. The word permutation, means change, and in mathematics, CHANGES IN THE ORDER IN WHICH THINGS SUCCEED EACH OTHER, ARE CALLED PERMUTATIONS. 1. What number of permutations can be made on the letters A B C ? If c be left out, A and B, as seen above, admit of 2 permutations. So, if в be left out, a and c admit of 2 permutations. And if a be left out, в and c admit of 2 permutations. But before each of these permutations, the letter left out may be placed; and as there were 2 permutations, 3 times, there are 6 in the whole. The pupil may make them for himself. From the above it will be seen that, of 2 things, there may be 1×2 =2 permutations; of 3, 1×2×3=6 permutations, and by the same mode of reasoning, it may be shown that, Sec. 105. PERMUTATION AND COMBINATION. 281 THE PERMUTATIONS, WHICH CAN BE MADE OF ANY NUMBER OF THINGS, ARE EQUAL TO THE CONTINUED PRODUCT OF THE NATURAL SERIES OF NUMBERS, FROM 1, UP TO THE NUMBER OF THINGS GIVEN. 2. Four gentlemen agreed to remain together, could arrange themselves differently at dinner. did they remain ? as long as they How many days A. 24 days. 3. 10 gentlemen made the same agreement, but they all died before it could be fulfilled. The last survivor lived 53 yrs. 98 days, after the agreement. How much did the bargain then want of being fulfilled, allowing 365 days to the year? A. 9,888 yrs. 237 d. 4. How many years will it take to ring all the possible changes on 12 bells, supposing that 10 can be rung in a minute, and that the year contains 365 d. 5 h. 49 m. ? A. 91 yrs. 26 d. 22 h. 41 m. 5. How many permutations may be made of the figures, 1, 2, 3, 4, 5, taken two at a time? Let 1 be placed by itself. To this, each other figure may be joined, making 4 permutations. Then 2 may be taken in the same way; and so with every other figure, there being 4 permutations each time. Then, as there are 5 figures, there will be 5X4-20 permutations of two figures. 6. How many permutations can be made on the figures above, taken three at a time? Here, if we set apart each arrangement of 2 figures, found as above, we may join to every one, each of the 3 remaining figures, which will make 3 times as many permutations. Now the permutations by twos, we have seen, are 5×4, and 3 times this number =5X4X3=60 permutations of three figures. By extending this mode of reasoning, we obtain the following. THE PERMUTATIONS WHICH CAN BE MADE OF ANY NUMBER OF THINGS, TAKEN A GIVEN NUMBER AT A TIME, ARE EQUAL TO THE CONTINUED PRODUCT OF A DECREASING NATURAL SERIES, WHOSE GREATEST TERM IS THE WHOLE NUMBER OF THINGS, AND WHOSE NUMBER OF TERMS, THE NUMBER TO BE TAKEN AT A TIME. 7. How many numbers can be expressed by the nine digits, taken four at a time? A. 3,024. 8. How many words of five letters each, may be made from an alphabet of 26 letters, supposing that a number of consonants may make a word? A. 7,893,600. From the letters, A, B and C, we can make three assemblages of two letters, of which no one shall contain exactly the same letters as another. These are A B, A C and в C. A combination means a collection of things, and in mathematics, COLLECTIONS OF WHICH NO TWO ARE EXACTLY ALIKE, CONSISTING EACH OF A GIVEN NUMBER OF THINGS, ARE CALLED COMBINATIONS. 9. How many combinations of two letters can be made from A B C D ? The permutations of two, we have seen to be 4X3=12. But on each combination of two, we have likewise seen, there can be made |