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COMPOUND INTEREST BY DECIMALS.

179

CASE 1. The principal, time, and rate given, to find the amount, or interest :

RULE. Multiply the principal by the ratio involved to the time, (found either by involution, or in Table II.) and the product will be the amount: from which subtract the principal, for the compound interest.

EXAMPLES. 1. What will 225 L. amount to in 3 years at 5 por

cont. per annum ?

1.05 X 1.05 X 1.05=1.157625 raised to the third power; then, 1.157625 X 225 = 260 L. 9s. 3d. 3 qrs. the Ans.

3 2. What will 480 L. amount to in 6 years, at 5 per cent. per annum?

Ans. 643L. 45. 11.0178d. 3. What is the amount of 500L. at 41 per cent. per ann. for 4 years?

Ans. 590 L,lls, 5d. 2.95+qrs. 4. What is the compound interest of a bond for 764 dollars for 4 years and 9 months, at 6 per cent. per annum?

Ans. 243 dols. 61 cts. +

CASE 2.

DISCOUNT, Or, the amount, rate, and time given, to find the principal:

RULE.
Divide the amount by the ratio involved to the time.

EXAMPLES. 1. What principal must be put to interest to amount to. 260 L. 9s. 3d. 3qrs. in 3 years, at 5 per cent. per annum?

260 L. 9s. 3d. 3qis. = 260.465625 L. 1.05 X 1.05 X 1.05 = 1.157625 ratio raised to the 3d power.

1.157625)260.465625(225 L. Ans. 2. What principal will amount to 547L. 9s. 10d. 2.0528 qrs. in 5 years, at 4 per cent. per annum? Ans. 450L.

3. What principal will amount to 619 L. 8s. 2d. 3.808qrs. in 4 years, at 54 per cent. ?

Ans. 500L.

ANNUITIES AT COMPOUND INTEREST. An annuity is a sum of money payable yearly, half yearly, or quarterly, for a number of years, during life, or for ever; and may draw interest if it remain unpaid after it becomes due. Tables to facilitate the calculations of annuities. TABLE III. Shewing the amount of L. 1 annuity.

surówno yr 4 per ct.42 per ct. 5 per ct. 52 per ct. 6 per ct. syr 14

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11.

1.
1.
1.

1. 2 2.04 2.045 2.05 2.055 2.06 3 3.1216 3.137025 3.1525 3.1682251 3.1836 4 4.246464 4.278191 4.310125 4.342266 4.374602 4 5 5.416322

5.47071 5.525631 5.581091 5.6370935 6! 6.632975 6.716892 6.801913 6.888051 6.9753181 6 71 7.898294 8.019152 8 142008 8.266894 8.393837 7 81 9.2 1226 9.380014: 9.549109 9.721573) 9.897468 8 9 10.582795 10.802114 11.026564 11.256259 11.4913169 10.12,006107) 12.28821 12.577892 12.875354 13.180795 10 1113.486351 13.841179| 14:206787| 14.583498 14.971643 11 1215.025805 15.464032 15.917126 16.38559 16.86994212 13 16.626838| 17.159913 17.712983) 18.286798 18.882138/13 14 18.291911 18.932109. 19.598632 20.292572 21.015066 14 15 20.023588 20.784054 21.578563 22.408663/ 23.275971|15 16 21.824531| 22.719337 23.657492 24.64114 | 25.672528 16 1723.697512 24.741707 25.840366, 26.996402 28.212881 17 18 25.645413 26.855084 28.132385 29.481205 30.90565318 1927.671229 29.065562 30.539004 32.102671 33.759993 19 20 29.778078 31.371423 33.065954 34.868918 36.785592 20 21 31.969202 33.783137 35.719252 37.786075 39.992728/21 22 54 247970 36.833378 38.505214 40.864309143.392291 22, 2336.617888 38.93703 41.430475 44 111846 46.995828/23 24 39.082604 41.639196 44.501999 47.537998 50.81557824 2541.645908 44.56521 47.727099. 51.152588 54.864513 25 2644.311745 47.5706451 51.113454 54.965979 59.15638326 27 47.684214 50.711324 51.669126! 58.989109 63.705766127 28 49 967632. 53.993:333 59.402583 63.23351 68.52811728 29 52.966286 57.423033| 62.322712 67.711355 73.639798 29 30 56.084938 61.007069 66.438847| 72.435478 79.058186 30 31;9.328335 64.7523881 70.76079 77.419429 84.801677|31 32162.701469 68.666245 75.298829 82.677498 90.88977832 3366.209527 72.75622680.063771 88.22476 97.343165 33 34/69.857904) 77.030256 85.066959 94.077122 104.18375434 3573.652225 31.496618 90.320307100.251363 111.434780/35 3677.598314 86.163966 95.836323 106.765188|119.12086736 3781.702246 91.041344 101.628139113.637274 127.268118 37 38.85.970336 96.138205/107709546120.887324 135.90420638 39 90.40915 101464424 114.095023128.536127145.058458 39 40105.125516107030329|120.799774136,605146/154.761966 40

ANNUITIES AT COMPOUND INTEREST.

181 TABLE IV. Shewing the present worth of L. 1 annuity for any num

ber of years, from 1 to 40.

.

yr 4 per ct. 4 per ct. 5 per ct. 5 per ct. 6 per ct yr

1 0.96154 0.956941 0.95231 0.94786 0.94339 is 2 1.83609 1.87267 1.85941 1.84632 1.83339 2

2.77509 2.74876 2.72325.2.69793 2.67301 3 4 3.62989) 3.58752 3.54595) 3.50514 3.4651 51 4.451821 4.38997 4.32988 4.270281 4.212365 61 5.24214) 5.15787 5.075691 4.99553 4.91732 6 71 6.402051 5.8927 5.78637 5.68297 5.58238 7 8) 6.73274 6.59589 6.463211 6.33457 6.20979 8 91 7.43533 7.26879 7.10782 6.95220 6.80169 9 101 8.1 1089 7.91272 7.72173 7.53762) 7.36008/10 11 8.76048 8.52892 8.3064 8.09254 7.8868711 12 9.385001 9.11858 8.863251 8:61852 8.3838412 13 9.985651 9.68285 9.393571 9.11708 8.85268|13 14|10.5631210.22282 9.89864. 9.58965) 9.2949814 15/11.41839 10.73954 10.37965 10.03759 9.7122515 1611.6522911.23401 10.83777 10.46216 9.10589-16 17.12.165671 1.7071911.27407 10.86461 10.47726 17 18 12.65929 12.15999 11.68958 11 24607 10.8276 118 1913.1339412.59329 12.08532 11.6076511.1581119 2013.5903213.00793 12.46221 11.9503411.46992/20 21 14.0291613.40472 12.8211512.27 524 11.76407/21 22 14.45111 13.7844213.163 12,5831712.04158/22 23 14.85684 14.14777 13.48857 12.87504|12.30338 23 24 15.24696 14.49548 13.7986,4 13,15170112.55035/24 2515.62208 14.82821 14.09394113.4139112.7833525 26|15.98277 15. i 4661|14.37518/13.6623013.0031626 27/16.32959 15.45130 14.6430313.8981013.2105327 28 16.66306 15.74287 14.8981314.1214213.4061628 29|16.9837 1 16.02189 15.14107 14,33310 13.59072/29 30 17.2920316.28889|15.37245 14.53375 13.7648330 31 17.58849 16.54439 15.5928114,723931 3.92908|31 32 17.87355 (6.78889 15.80268|14.90420 14.08404132 3318.14764 17.02286 16.00255 15.07 507 14.23023133 34118.41126/17.24676 16.1929 |15.23703|14.3681434 35 18.6646117.46101/16.37419|15.39055|14.49825 35 36 18.90828117.66604 16,54685 15.53607|1 4.62098 36 37 19.14258 17.86224 16.7112915,67400 14.73678 37 38|19.3678618.0499916.86789|15.80474 14.8460238 39 19.5844818.22965 17.01704 15.92866 14.94907139 40 19.79277|18.40ļ58 17.15909/160461214.92E40 40

TABLE V.

Rate Jhalf yearlys quarterly

The construction of this table, per cent. payinents. payments.

is from an algebraic theorem given

by the learned A. De Moivre, in 3 1.007 445 1.011181

his treatise of annuities, on lives,

which may be in words, thus : 31 1.008675 1.013031

For half yearly payments take a 1.009902 1.014877

unit from the ratio, and from the 41 1.011126/1.016720

square root of the ratio ; half the 5 1.0123481.018559 quotient of the first remainder di. 51 11.0135671.020395

vided by the latter, will be the ta6 1.014781 1.022257

bular number.

For quarterly payments use the 64 11.015993/1.024055

4th root as above, and take one 7 11.017204!1.025880 quarter of the quotient.

CASE 1. The annuity, time, and rate of interest given, to find the amount:

RULE. From the ratio involved to the time take an unit or one, for the dividend; which divide by the ratio less one; and multiply the quotient by the annuity for the amount or answer. Or, by Table III.

Multiply the number under the rate, and opposite to the time, by the annuity, and the product will be the amount for yearly payments.

If the payments be half yearly or quarterly, the amount for the given time, found as above, multiplied by the proper number in Table V. will be the true amount.

EXAMPLES.

1. What will an annuity of 50 L. per annum, payable. yearly, amount to in 4 years, at 5 per cent. ?

1.05 X 1.05 X 1.05 X 1.05 15,21550625
1.05—1=.05).21550625

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4.310125

50

Ańs, L. 215.506250 =213 L. 10s. 1d. 2grs.

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ANNUITIES AT COMPOUND INTÉREST.

183

2. What will an annuity of 30 L. per annum, payable yearly, amount to in 4 years, at 5 per cent. per annum, and what would be the respective amounts if the payments were to be half yearly or quarterly?

F Amount for yearly payments is L. 129.50375. Ans. for half yearly

L. 130.9004. for quarterly

L. 131.7035. 3. If a salary of 35 L. per annum to be paid yearly, be omitted for 6 years at 51 per cent., what is the amount?

Ans. 241 L. Is. 70. 2.5+qrs.

CASE 2.

The annuity, time, and rate given, to find the present worth:

RULE.

Divide the annuity by the ratio involved to the time, and subtract the quotient from the annuity; divide the remainder by the ratio less one, and the quotient will be the present worth : Or, by Table IV.

Multiply the number under the rate, and opposite the time by the annuity, and the product will be the present worth.

When the payments are half yearly or quarterly, mul, tiply the present worth so found, by the proper number in Table V.

EXAMPLES.

1. What is the present worth of a pension of 30 L. per annum for 5 years, at 4 per cent. ? Ans. 133L. Ils. Id. Number from Table IV. 4.45132

X30 annuity.

L. 133.55460

Or, 133 L. Ils. 1.104d. 2. What is the present worth of 201.. a year for 6 years, payable either yearly, half yearly, or quarterly, computing at 5 per cent. per annum?

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