5. What sum of ready money is equivalent to an annuity of $200, to continue 3 years, at 4 per cent.? A. $556,063. 6. What is the present worth of an annual salary of $800, to continue 2 years?-1469001. 3 years ?-2146967. 5 years?-3407512. A. $7023,48. ANNUITIES AT COMPOUND INTEREST. XCI. The amount of an annuity, at simple and compound interest, is the same, excepting the difference in interest. Hence, To find the Amount of an Annuity at Compound Interest; Proceed as in ¶ XC., reckoning compound, instead of simple interest. 1. What will a salary of $200 amount to, which has remained unpaid for 3 years? $224,72 The amount of $200 for 2 years = The 3d year.. A. $636,72 2. If the annual rent of a house, which is $150, remain in arrears for 3 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 period of years, would be tedious. This trouble will be prevented, by finding the amount of $1, or £1, annuity, at compound interest, for a number of years, as in the following TABLE 1. Showing the amount of $1 or £1 annuity, at 6 per cent. compound in terest, for any number of years, from 1 to 50. Yrs. 6 per ct. Yrs.16 per ct.Yrs.16 per ct.Yrs.16 per ct. Yrs.16 per ct. 1 1,0000 11 14,9716 21 39,9927 31 2 2,0600 12 16,8699 22 43,3922 32 3 3,1836 13 18,8821 23 46,9958|| 33 84,8016 41 165,0467 90,8897 42 175,9495 97,3431 43 187,5064 104,1837 44 199,7568 2554,8645 35 111,4347 45 212,7423 26 59,1563 36 119,1208 46 226,5068 4 4.3746 14 21,0150 24 50,8155 34 7 8,3938 17 28,2123 27 63,7057 37 127,2681 47 231,0972 8 9,8974 18 30,9056 28 68,5281 38 135,9042 48 245,9630 9 11,4913 10 1933,7599 29 73,6397 39 145,0584 49 261,7208 40 154,7619 50 278,4241 13,18072036,78553079,0581 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 Table, the amount of $1, at the given rate and time, 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 X 120 = $2793,108, Ans. 4. What will be the amount of an annual salary of $400, which has been in arrears 2 years?-894. 3 years?-127344. 4 years?-174984. 6 years?-279012. 12 years?-674796. 20 years?-147142. Ans. $28099,56. 5. If you lay up $100 a year, from the time you are 21 years of age till yon are 70, what will be the amount 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 compo interest, would amount to as much as the amount of the given annuity the three years. Finding the amount of $120 by the Table, as before, w have $382,032; then, if we divide $382,032 by the amount of $1, compoun 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, $382,032. 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. Nute. 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. $842,472. The operations in this rule may be much shortened 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 interest, for any number of years, from 1 to 32. Years. 6 per cent. Years. 6 per cent. Years. 6 per cent. Years. 6 per cent. To find the present worth of any annuity, by this Table, we have only to multiply the present worth of $1, found in the Table, by the given annuity, and the product will be the present worth required. 8. What sum of ready money will purchase an annuity of $300, to continue 10 years? The present worth of $1 annuity, by the Table, for 10 years, is $7,36008; then 7,36008 x 300 = $2208,024, Ans. 9. What is the present worth of a yearly pension of $60, to continue 2 years?-1100034. 3 years?-1603806. 4 years?-207906. 20 years?-6881952. 30 years?-8258898. A. $2364,9624. 8 years?-3725874. 10. What salary, to continue 10 years, will $2208,024 purchase? This example is the 8th example reversed; consequently, $2208,024 + 7,36000300, the annuity required. A. $300. Hence, To find that Annuity which any given Sum 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 annuity required. 11. What salary, to continue 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 four 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 $27000 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 year, $7791,97032 2d $6934,825, .... 4th.. $6171,970, Proof, $26999,998 +. 2 3 $7791,97032 4.... $7791,97032 For the entire work of the last example, see the Key. PERMUTATION. ¶ XCII. PERMUTATION is the method of finding how many different ways any number of things may be changed. 1. How many changes may be made of the three first letters of the alphabet? In this example, had there been but two letters, they could only be changed twice; that is, a, b, and b, a; that is, 1 x 2 = 2; but, as there are three letters, they may be changed 1 x 2 x 36 times, as follows: 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 1 up to the given number, and the last product will be the number of changes required. 2. How many different ways may the first five letters of the alphabet be arranged? A. 120. 3. How many changes may be rung on 15 bells? be rung, allowing 3 seconds to every round? 3923023104000 seconds. and in what time may they A. 1307674368000 changes; 4. What time will it require for 10 boarders to seat themselves differently every day at dinner, allowing 365 days to the year? 4. 994138 years. 5. Of how many variations will the 26 letters of the alphabet admit? A. 403291461126605635584000000. POSITION Is a rule which teaches, by the use of supposed numbers, to find true ones. It is divided into two parts, called Single and Double. SINGLE POSITION. ¶ XCIII. This rule 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 half as many more, one third, and one fourth as many more, I should have 296." How many had he? Let us suppose he had 24 as many We have now found that we did not suppose have been 296. But 24 has been increased in the the right number. If we had, the amount would 12 | same manner to amount to 74, that some unknown 8 number, the true number of scholars, must be, to amount to 296. Consequently, it is obvious, that 674 has the same ratio to 296 that 24 has to the true number. The question may, therefore, be solved 74 by the following statement: As 74 296: 24: 96, Ans. This answer we prove to be right by increasing it by itself, one half itself, one third itself, and one fourth itself; 96 96 48 Suppose any number you choose, and proceed with it in the same manner you would with the answer, to see if it were right. Then say, As this result: the result in the question :: the supposed number: number sought. More Exercises for the Slate. 2. James lent William a sum of money on interest, and in 10 years it amounted to $1600; what was the sum lent? A. $1000. 3. Three merchants gained, by trading, $1920, of which A took a certain sum, B took three times as much as A, and C four times as much as B; what share of the gain had each? A. A's share was $120; B's, $360; and C's, $1440. 4. A person having about him a certain number of crowns, said, if a third, a fourth, and a sixth of them were added together, the sum would be 45; how many crowns had he? A. 60. 5. What is the age of a person, who says, that if of the years he has lived be multiplied by 7, and of them be added to the product, the sum would be 292? A. 60 years. 6. What number is that, which, being multiplied by 7, and the product divided by 6, the quotient will be 14? A. 12. 23 * |