2. When the annuity is not paid at the time it becomes due, it is said to be in arrears. 3. 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. 4. Hence, to find the amount of an annuity-Calculate the interest on each 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. 5. If the annual rent of a house, which is $200, remain unpaid (that is, in arrears) 8 years, what is the amount? 6. 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. The amount of $200 for 7 years=$284 The amount of $200 for 6 years=$272 $1,936 A. 7. If a man, having an annual pension of $60, receive no part of it till the expiration of 8 years, what is the amount then due? A. $580.80. 8. What would an annual salary of $600 amount to, which remains unpaid (or in arrears) for 2 years ?-1,236. For 3 years?-1,908. For 4 years?-2,616. For 7 years?-4,956. For 8 years?-5,808. For 10 years?-7,620. A. Total, $24,144. 9. What is the present worth of an annuity of $600, to continue 4 years? The present worth [LXXXIII.] is such a sum as, if put at interest, would amount to the given annuity; hence, $600-$1.06 $566.037, present worth, 1st year. 10. 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. 11. What sum of ready money is equivalent to an annuity of $200, to continue years, at 4 per cent.? A. $556.063. 12. What is the present worth of an annual salary of $800, to continue 2 years?-1,469.001. 3 years?-2,146.967? 5 years?— 3,407.512. A. Total, $7,023.48. Q. What is meant by arrears and amount? 2, 3. Rule for finding the amount? 4. For finding the present worth? 10. ANNUITIES AT COMPOUND INTEREST CVII. 1. The amount of an annuity at simple and compound interest is the same, excepting the difference in interest. 2. Hence, to find the amount of an annuity at compound interest.-Proceed as in cvi., reckoning compound instead of simple interest. 3. What will a salary of $200 amount to, which has remained un· paid for 3 years? The amount of $200 for 2 years=$224.72 The amount of $200 for 1 year = $212.00 4. 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. 5. Calculating the amount of the annuities in this manner, for a long period of years, would be tedious. This trouble will be pre vented, by finding the amount of $1, or £1, annuity, at compound in terest, for a number of years, as in the following TABLE I. Showing the amount of $1, or £1, annuity, at 6 per cent., compound interest, for any number of years, from 1 to 50. Y.16 per cent. Y. 6 per cent. Y. 6 per cent. 21 39.9927 31 84.8016 41 165.0467 22 43.3922 32 90.8897 42 175.9495 23 46.9958 33 97.3431 43 187.5064 34 104.1837 44 199.7568 Y.(6 per cent.[ Y.6 per cent. 7 8.3938 8 9.8974 15 23.2759 25 54.8645 36 119.1208 46 226.5068 27 63.7057 37 127.2681 47 231.0972 6 6.975316 25.6725 26 59.1563 17 28.2123 1830.9056 28 68.5281 38 135.9042 48 245.9630 9 11.4913 19 33.7599 29 73.6397 39 145.0584 49 261.7208 10 13.1807 20 36.7855 30 79.0581 40 154.7619 50/278.4241 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. 6. 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 re quired. 7. 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 × 120=$2,793.108. A. 8. What will be the amount of an annual salary of $400, which has been in arrears 2 years ?-824. 3 years?-1,273.44. 4 years?— CVII. Q. What is meant by the amount of an annuity at compound interest? 1. How is it found? 2. 1,749.84. 6 years?-2,790.12. 12 years?-6,747.96. 20 years ?14,714.2. A. Total, $28,099.56. 9. If you lay up $100 a year, from the time you are 21 years of age till you are 70, what will be the amount at compound interest? A. $26,172.08. 10. What is the present worth of an annual pension of $120, which is to continue 3 years? 11. 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 three 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 $1, 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, $382.032. The amount of $1 for 3 years at compound interest is $1.19101; then, $382.032÷$1.19101= $320.763, A. 12. 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. 13. NOTE.-The amount of $1 may be found, ready calculated, in the table of compound interest, [LXXXII.] 14. What is the present worth of an annual rent of $200, to continue 5 years? A. $842.472. 15. 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. 8 6.20979 1610.10589 24 12.55035 32 14.08398 16. 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. 17. 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 × 300-$2,208.024. A. Q. How is the present worth found? 12. 18. What is the present worth of a yearly pension of $60, to con tinue 2 years?-110.0034. 3 years?—160.3806. 4 years?—207.906 8 years?-372.5874. 20 years?-688.1952. 30 years?-825.8898 Total, $2,364.9624. 19. What salary, to continue 10 years, will $2,208.024 purchase? This example is the 17th example reversed; consequently, $2.208.024 ÷7.36008=300, the annuity required. A. $300. 20. 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 1.; the quotient will be the annuity required. 21. What salary, to continue 20 years, will $688.195 purchase? A. $60+. 22. What annuity, to continue 10 years, is equivalent to $3,680.041 A. $500. 23. To divide any sum of money into annual payments, which, when due, shall form an equal amount at compound interest.—First find an equivalent annuity as above, (20,) then its present worth for each required period of time. (15.) 24. A certain manufacturing establishment in Massachusetts was actually sold for $27,000, and the sum 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 $27,000; now, what were the principals of said notes ?* The first note is $7,350.915; amount for 1 year, $7,791.97032. PERMUTATION. CVIII. 1. PERMUTATION is the method of finding how many different ways any number of things may be changed. 2. How many changes may be made of the first three 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, 1×2=2; but, as there are three letters, they may be changed 1×2×3=0 times, as follows-1, a, b, c; 2, a, c, b; 3, b, a, c; 4, b, c, a; 5, c, b, a; 6, c, a, b. Q. What is the rule for finding what sum a given annuity will purchase? 20. CVIII. Q. What is Permutation? 1. How many changes can be made with the first three letters of the alphabet? 2. What are they? 2. Rule? 3. *The annuity which $27,000 will purchase, found as before, is 7,791.97032+. To obtain an exact result, we must reckon the decimals, which were rejected in forming the tables. This makes the last divisor 3.4651056. 3. 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. 4. How many different ways may the first five letters of the alphabet be arranged? A. 120. 5. How many changes may be rung on 15 bells, and in what time may they be rung, allowing 3 seconds to every round? A. 1,307,674,368,000 changes; 3,923,023,104,000 seconds. 6. What time will it require for 10 boarders to seat themselves differently every day at dinner, allowing 365 days to the year? A. 9,94133 years. 7. Of how many variations will the 26 letters of the alphabet admit? A. 403,291,461,126,605,635,584,000,000. POSITION. 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. CIX. 1. SINGLE POSITION teaches to resolve those questions whose results are proportional to their suppositions. 2. 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? Suppose he had 24 as many 12 as many 8 6 74 We have now found that we did not suppose 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 is 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, A. 3. This answer we prove to be right by increasing it by itself, one half of itself, one third of itself, and one fourth of itself, as, 96+96+ 48+32+24= 296. RULE. 4. 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. CIX. Q. What is Position? Single Position? 1. Rule? 4. |