ANNUITIES AT SIMPLE INTEREST. T XC. An annuity is a sum of money, payable every year, for acontain number of years, or forever. When the annuity is not paid at the time it becomes due, it is said to be in arrears. The sum of all the annuities, such as rents, pensions, &c., remaining unpaid, with the interest on each, for the time it has been due, is called the amount of the annuity. Hence, to find the amount of an annuity; Calculate the interest on cach annuity, for the time it has remained unpaid, and find its amount: then the sum of all these several amounts will be the amount required. 1. If the annual rent of a house, which is $200, remain unpaid, (that is, in arrears,) 8 years, what is the amount? In this example, the rent of the last (8th) year being paid when due of course, there is no interest to be calculated on that year's rent. $1336, Ans 2. If a men, having an annual pension of $60, receive no part of it till the expiration of 8 years, what is the amount then due? A. $560,80. 3. What would an annual salary of $600 amount to, which remains unpaid (or in arrears) for 2 years? (1236) For 3 years? (1908) For 4 years? (2616) For 7 years? (4956) For 8 years? (5208) For 10 years? (7620) Ans. $24144 4. What is the present worth of an annuity of $600, to continue 4 years? The prosent worth, (T LXVII.,) is such a sum as, put at interest, would a:nount to the given annuity; hence, 11 if $566,037, present worth, lat year. $600 $1,06 ་ $600 $1,12= $535,714, $600 $1,18 =$508,474, $600 $1,24 $483,870, 2d 3d ....... Ans., $2094,095, present worth required. Hence, to find the present worth of an annuity;- Find the present worth of each year by itself, discounting from the time it becomes due, and the sum of all these present worths will be the answer. 5. What sum of ready money is equivalent to an annuity of $200, to com tinne 3 years, at 4 per cent.? A. $556,063. 6. What is the present worth of an annual salary of $800, to continue years (1469001) 3 years? (2146967) 5 years? (3407512) A. $7023. ANNUITIES AT COMPOUND INTEREST. TXCI. The amount of an annuity, at simple and compound interest, i the same, oxcepting the difference in interest. Hence, to find the amount of an annuity at com pound interest;— Proceed as in ¶ XC., reckoning compound, instead of simpl、 interest. 1. What will a salary of $200 amount to, which has remained unpaid for S ycara? The amount of $200 for 2 years = $224,79 The 3d year, ・ $212,00 $200,00 A. $636,72 2. If the annual rent of a house, which is $150, remain in arrears for years, what will be the amount due for that time? A. $477,54. Calculating the amount of the annuities in this manner, for a long perio of years, would be tedious. This trouble will be prevented, by finding th amount of $1, or 1£, annuity, at compound interest, for a number of yeari as in the following TABLE I. Showing the amount of $1 or 1 annuity, at 6 per cent. compound interes for any number of years, from 1 to 50. 84,8016 41 165,0-467 Yrs.16 per cent. Yrs.16 per cent. Yrs. 6 per cent.fi Yrs. 16 per cent. Yrs. 16 per cent. 4,3746 14 21,0150 24 50,8155 33 97,3431|| 43 187,5064 34 104,1837 44 199,7568 4 6 6,9753 16 25,6725 26 59,1563 36 119,1208 46 226,5068 8 63,7057 37 127,2681 47 231,0972 68,5281 38 125,9042 48 215,9630 9 11,4913 19 33,7599 29 73,6397 39 145,0534 49 261,7208 It is evident, that the amount of $2 annuity is 2 times as much as one of $1 and one of $3,3 times as much; hence, To find the amount of an annuity, at 6 per cent. ;— Find by the Tuble the amount of $1, at the given rate and tune, and multiply it by the given annuity, and the product will be the amount required. 3 What is the amount of an annuity of $120, which has remained unpaid 15 years? The amount of $1, by the Table, we find to be $23,2759; therefore, $23,2759. *1·20== $2793,108, Ans. What will be the amount of an annual salary of $400, which has been in ariears 2 years? (824) 3 years? (127344) 4 years? (174984) 6 years. 279012) 12 years? (674796) 20 years? (147142) Ans. $23099,56. 5. If you lay up $100 a year from the time you are 21 years of ago till you are 70, what will be the aniount at compound interest? A. $26172,08. 6. What is the present worth of an annual pension of $120, which is to continue 3 years? In this example, the present worth is evidently that sum, which, at compound interest, would amount to as much as the amount of the given annuity for the 3 years? Finding the amount of $120 by the Table, as before we have $382,032; then, if we divide $382,032 by the amount of $i, compound interest, for 3 years, the quotient will be the present worth. This is evident From the fact, that the quotient, multiplied by the amount of $1, will give the amount of $120, or, in other words, $332,032. The amount of $1 for 3 years at compound interest, is $1,19101; then, $382,032 $1,19101 = $320,763, Ans. Hence, to find the present worth of an annuity; Find its amount in arrears for the whole time; this amount, divided by the amount of $1 for said time, will be the present worth required. Note. The amount of $1 may be found ready calculated in the Table of compound interest, ¶ LXXI. 7. What is the present worth of an annual rent of $200, to continue 5 years! A. $842,172. The operations in this rule may be muchi ahortened by calculating the present worth of $1 for a number of years, as in the following TABLE II. Showing the present worth of $1 or 1 annuity, at 6 per cent. compound in terest for any number of years, from 1 to 32. To find the present worth of any annuity, by this Table, we have only to multiply the present worth of $1, found in the Ta ble, by the given unnuity, and the product will be the present worth required. 8. What sum of ready money will purchase an annuity of $300, to contin e 10 years? The present worth of $1 annuity, by the Table, for 10 years, is $7,36008; then 7,30008 X 300 $2208,024, Ans. 9. What is the prosent worth of a yearly pension of $60, to continue 2 years? (1100034) 3 years? (1603806) 4 years? (207906) 8 years? (3725874) 20 years? (6881952) 30 years? (8258808) A. $2364,9624. 10. What salary, to continuo 10 years, will $2208,024 purchase? This example is the 8th example reversed; consequently, $2208,0247,36000 300, the annuity required. .9. $300. Hence, to find that annuity which any given sun will purchase ; Divide the given sum by the present worth of $1 annuity for the given time, found by Table II., the quotient will be the annui ty required. 11. What salary, to contitue 20 years, will $688,95 purchase? A $60-+ To divide any sum of money into annual payments, which, when due, shall form an equal amount, at compound interest; 12. A certain manufacturing establishment, in Massachusetts, was actually sold for $27000, which was divided into 4 notes, payable annually, so that the principal and interest of each, when due, should form an equal amount, at compound interest, and the several principals, when added together, should make $27000; now, what were the principals of said notes? It is plain, that, in this example, if we find an annuity to continue 4 years, which $27000 will purchase, the present worth of this annuity for 1 year will be the first payment, or principal of the note; the present worth for 2 years, the second, and so on to the last year. The annuity which $7000 will purchase, found as before, is 7791,97032+. Note. To obtain an exact result, we must reckon the decimals, which were rejected in forming the tables. This makes the last divisor 3,4651056. The 1st is $7350,915, amount for 1 yr. $7791,97032 $6934,825, 2d 2 $7791,97039 ...... 4th $6171,970, Proof, $26999,998+ .... PERMUTATION. TXCII. PERMUTATION is the method of finding how many differen ways any number of things may be changed. 1. How many changes may be made of the three first letters of the al phabet? In this example, had there been but two letters, they could only be changed twice; that is, a, b, and b, a; that is, 1X22; but, as there are three lot ters, they may be changed 1X2 X 36 times, as follows: (a, b, c. 2a, c, b. 3 b, a, c. 4) b, c, 8. c, b, a. 6 [c, a, b. Hence, to find the number of different changes or permutations, which may be made with any given number of different things; Multiply together all the terms of the natural series, from I up to the given number, and the last product will be the number of changes required. 2. How many different ways may the first 5 letters of the alphabet be arranged? A. 120. 3. How many changes may be rung on 15 bells, and in what time may they be rung, allowing 3 seconds to every round? A. 1307674368000 changes; C323023104000 seconds. 4. What time will it require for 10 boarders to seat themselves differently ev ery day at dinner, allowing 365 days to the year? A. 9941 8 years. 5. Of how many variations will tho 26 letters of the alphabet admit? A. 403291461126605635584000000 POSITION s a rule which teaches, by the uro of supposed numbers, to find true ones. It s divided into two parts, called Singlo and Double. SINGLE POSITION. 'TXCIII. This rulo teaches to resolve those questions whose results are proportional to their suppositions. 1. A schoolmaster, being asked how many scholars he had, replied, "If I had as many more as I now have, one balf as many more, one third, and one ourth as many more, I should have 296." How many had he? Let us suppose he had 24 Then as many more=24 as many 12 71 We have now found that we did not supposo the right number. If we had, the amount would have been 296. But 24 has been increased in the same manner to amount to 74, that some unknown number, the true number of scholars, must be, to amount to 296. Consequently, it 18 obvious, that 74 has the same ratio to 296 that 24 has to the true number. The question may, therefore, be solved by the following statement: As 74: 296:: 24: 96, Ans. This answer we prove to be right by increasing it by itself, one half ( 96 itself, one third itself, and one fourth itself; 96 48 Thus, 32 24 296 From these illustrations we derive the following RULE. Suppose any number you choose, and proceed with it in he seme manner you would with the answer, to see if it were right. II. Then say, As this result: the result in the question : : the supposed number: number sought. 2 More Exercises for the Slate. ames lent William as am of money on interest, and in 10 years it amount ed to $1000; what was the sum lent? A. $1000. |