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TABLE,

SHOWING THE PRESENT WORTH OF AX AXXUITY OF OXE DOLLAR PER

ANNUM, TO CONTINUE FOR ANY NUMBER OF YEARS NOT EXCEEDING FIFTY.

3 per cent. 3} per cent. 4 per cent.

5 per cent.

6 per cent.

7 per cent.

Years.

10 OTA CON Years.

1 0.970 874 0.966 184 0.961 538 0.932 381 0.943 396 0.934 579 1 2 1.913 470 1.899 694 1.886 095 1.859 410 1.833 393 1.808 017 2 3 2.828 611 2.801 637 2.775 091 2.723 218 2.673 012 2.624 314 3 4 3.717 098 3.673 079 3.629 895 3.545 951 3.465 106 3.387 209 4 5 4.579 707 4.515 052 4.451 822 4.329 477 4.212 364 4.100 195 5 6 5.417 191 5.328 553 5.242 137) 5.075 692 4.917 324 4.766 537 6 7 6.230 283 6.114 544 6.002 055 5.786 373 5.582 381 5.389 286 8 7.019 692 6.873 956 6.732 745 6.463 213 6.209 744 5.971 295 8 9 7.786 109 7.607 687 7.435 332 7.107 822 6.801 692 6.515 228 9 10 8.530 203 8.316 605 8.110 896 7.721 735 7.360 087 7.023 577 10 11 9.252 624 9.001 551 8.760 477) 8.306 414 7.886 875 7.498 669 11 12 9.954 004 9.663 334 9.38 074 8.863 252 8.383 844 7.942 671 12 13 10.634 955 10.302 738) 9.985 648 9.393 573 8.852 683 8.357 635 13 14 11.296 073 10.920 520 10.563 123 9.898 641 9.294 984 8.745 452 14 15 11.937 935 11.517 411 11.118 387 10.379 658 9.712 249 9.107 898 15 16 12.561 102 12.094 117 11.652 296 10.837 770 10.105 895 9.446 632 16 17 13.166 118 12.651 321 12.165 669 11.274 066 10.477 260 9.763 206 17 19 13.753 513 13.189 682 12.659 297 11.689 587 10.827 603 10.059 070 18 19 14.323 799 13.709 837 13.133 939 12.085 321 11.158 116 10.335 578 19 20 14.877 475 14.212 403 13.590 326 12.462 210 11.469 421 10.593 997 20 21 15.415 024 14.697 974 14.029 160 12.821 153 11.764 077 10.835 527 21 22 15.936 917 15.167 125 14.451 115 13.163 003 12.041 582 11.061 241 22 23 16.443 608 15.620 410 14.856 842 13.488 574 12.303 379 11.272 187 23 24 16.935 542 16.058 368 15.246 963 13.798 642 12.550 358 11.469 334 24 25 17.413 148 16.481 515 15.622 080 14.093 945 12.783 356 11.653 583 25 26 17.876 842 16.890 352 15.982 769 14.275 185 13.003 166 11.825 779 26 27 18.327 031 17.285 365 16.329 586 14.643 034 13.210 534 11.986 709 27 28 18.764 108 17.667 019 16.663 063 14.898 127 13.406 164 12.137 111 28 29 19.188 455 18.035 767 16.983 715 15.141 074 13.590 721 12.277 674 29 30 19.600 441 18.392 015 17.292 033 15.372 451 13.764 831 12.409 041 30 31 20.000 428 18.736 276 17.588 494 15.592 811 13.929 086 12.531 814 31 32 20.338 766 19.068 865 17.873 552 15.802 677 14.084 043 12.646 555 32 33 20.765 792 19 390 208 18.147 646 16.002 549 14.230 230 12.753 790 33 34 21.131 837 19.700 684 18.411 198 16.192 204 14.368 141 12.854 009 34 35 21.487 220 20.000 661 18.661 613 16.374 194 14.498 246 12.947 672 35 36 21.832 252 20.290 494 18.908 282 16.546 852 14.620 987 13.035 208 36 37 22.167 235 20.570 525 19.142 579 16.711 287 14.736 780 13.117 017 37 38 22.492 462 20.841 087 19.367 864 16.867 893 14.846 019 13,193 473 38 39 22.808 215 21.102 500 19.584 485 17.017 04114.949 075 12.264 928 39 40 23.114 772 21.355 072 19.792 774 17.159 086 15.046 297 13.331 709 40 41 |23.412 400 21.599 104 19.993 052 17.294 368 15.138 01613.394 120 41 42 23.701 359 21.834 883 20.185 627 17.423 208 15.224 543 13.452 449 42 43 23.981 902 22.062 689 20.370 795 17.545 912 15.306 173 13.506 962 43 44 24.254 274 22.282 791 20.548 841 17.662 773 15.383 182 13.557 908 44 45 24.518 71322.495 450 20.720 040 17.774 070'15.455 832 13.605 522 45 46 24.775 449 22.700 918 20.884 654 17.880 067 15.524 370 13.650 020 46 47 25.024 708 22.899 438 21.012 93617.981 016 15.589 02813.691 608 47 48 25.266 707 23.091 244/21.195 131 18.077 158 15.650 027 13.730 474 48 49 25.501 657 23.276 567 21.341 472 18.168 722 15.707 572 13.766 799 49 50 25 729 76423.455 618 21.482 185 18.255 925 15.761 861 13.800 746 50

OPERATION.

Or,

574. To find the amount of an annuity, at compound interest, forborne, or in arrears, for any number of years.

Ex. 1. What will an annuity of $ 60, unpaid, or in arrears, 4 years, amount to, at 6 per cent. compound interest?

Ans. $ 262.476. The amounts of the several

payments form a geometrical 1.064 1

series, of which the annuity is x 60 = 262.476. 1.06 1

the first term, the amount of

$ 1 for one year the ratio, the Or, 4.374616 X 60 262.476.

years

the number of the terms, and the amount required is the

sum of the series. Hence, RULE. — Find the sum of the series as in geometrical progression.

Multiply the amount of $ 1 for the given time, found in the table, by the annuity, and the product will be the required amount.

NOTE. — The amount of an annuity at simple interest corresponds to the bum of an arithmetical series, of which the annuity is the first term, the interest on the annuity for one term the common difference, and the time in years the number of terms. 2. What will an annuity of $ 500 amount to for 5 years,

at 6 per cent. compound interest ?

Ans. $ 2818.546. 3. What is the amount of an annuity of $ 80, unpaid, or in arrears, for 9 years, at 5 per cent. compound interest ?

Ans. $ 882.125. 4. What is the amount of an annuity of $ 1000, forborne for 15 years, at 31 per cent. compound interest ?

Ans. $ 19295.68. 5. What will an annuity of $30, payable semiannually, amount to, in arrears for 3 years, at 7 per cent. compound interest?

6. Suppose a salary of $ 600 per year, payable quarterly, to remain unpaid 51 years; to what sum will it amount, at 6 per cent. compound interest ?

Ans. $ 3875.63. 575. To find the present worth of an annuity, at compound interest.

Ex. 1. What is the present worth of an annuity of $ 60, to be continued 4 years, at 6 per cent. compound interest?

Ans. $ 207.906.

OPERATION,

The present worth re$3.465106 X 60 = $ 207.906+quired evidently may be ob

tained by finding the amount of the given annuity, by the last articles, and then finding in the usual way the present worth of that amount. A more expeditious method, however, is to find, in the table, the present worth of an annuity of $ 1 for the given time and rate, and take that sum as many times as there are dollars in the given annuity, as in the operation.

RULE. Multiply the present worth of an annuity of $1 for the given time and rate by the number denoting the given annuity.

2. What is the present worth of an annuity of $100, for 9 years, at 6 per cent. ?

Ans. $ 680.169. 3. What is the present worth of an annuity of $ 200, for 7 years, at 5 per cent. ?

Ans. $ 1157.27. 4. Required the present worth of an annuity of $ 500, to continue 40 years, at 7 per cent.

5. A gentleman wishes to purchase an annuity, which shall afford him, at 6 per cent. compound interest, $ 500 a year, for ten years. What sum must he deposit in the annuity office to produce it?

Ans. $ 3680.04. 6. If a widow be entitled to $ 160 a year, payable semiannually, from a fund, for 8 years, what is its value at present, at 6 per cent. compound interest ?

Ans. $ 1004.88. 576. To find the present worth of an annuity in perpetuity.

Ex. 1. What is the present worth of a perpetual lease, which yields an income of $ 600, the rate of interest being that of 6 per cent. ?

Ans. $ 10000. The question is evidently the same

as one requiring what principal in one $ 600 ; .06 = $ 10000.

year, at 6 per cent. interest, will yield

$ 600. RULE. Divide the given annuity by the number denoting the interest of $ 1 for one year.

NOTE. — When the annuity is payable quarterly, semiannually, or in any other periods less than a whole year, the annuity must be increased by the interest which may thus accrue on the parts of the annuity payable before the end of the year, before dividing by the interest of $1 for one year.

2. A ground rent in the city of Philadelphia yields an annual income of $ 963, at 6 per cent. interest. What is the value of the estate ?

Ans. $ 16050.

OPERATION,

OPERATION.

3. What is the present value of a perpetual lease, yielding an income of $ 6335, interest being at 7 per cent.

Ans. $ 90500. 4. What sum should be paid for a perpetual annuity of $ 1200, payable semi-annually, interest being at 5 per cent. ?

Ans. $ 24000. 577., To find the present worth of an annuity in reversion.

Ex. 1. What is the present worth of an annuity of $ 300, to commence in 3 years, and to continue 5 years, allowing compound interest at 6 per cent. ?

Ans. $ 1061.03.

The present worth $ 6.209744 $ 2.673012 = $ 3.536732; of an annuity of $1, $ 3.53673 X 300 = $ 1061.03. at 6 per cent., com

mencing at once, and continuing till the termination of the annuity, or for 3 + 5 = 8 years, is $ 6.209744; and the present worth of the same annuity up to the time of the commencement of the reversion is $ 2.673012. The difference of these present worths multiplied by the number of dollars in the given annuity is the present worth of the reversion.

RULE. — Find the present worth of an annuity of $ 1, commencing immediately and continuing till the reversion COMMENCES, and also till the reversion TERMINATES; and multiply the difference of these present worths by the number of dollars in the given annuity. The result will be the present worth required.

2. The reversion of a lease of $ 350 per annum, to continue 11 years, which commences 9 years hence, is to be sold. What is its worth, allowing the purchaser 6 per cent. per annum for his ready money?

Ans. $ 1633.70. 3. A father presents to his daughter, for 8 years, a rental of $ 70 per annum, payable yearly, and its reversion for the 12 years succeeding to his son. What is the present value of the gift to his son, allowing 4 per cent. compound interest?

Ans. $ 480.03. 4. What is the present worth of the reversion of a perpetuity of $ 240 per annum, payable yearly, but not to come into possession till the expiration of 100 years, compound interest being allowed at 6

per
cent. ?

Ans. $ 11.78. 578. To find the annuity, the present worth, time, and rate being given.

OPERATIOX.

Ex. 1. What annuity, continued for 4 years, at 6 per cent. compound interest, is now worth $ 207.90 ?

Ans. $ 60. The present value represented by

the debt, divided by the present $ 207.90 • 3.465 = $ 60. worth of $ 1 for the given time and

rate, gives the annuity required. RULE. - Diviile the given present worth by the present worth of an annuity of $ 1 for the given time and rate, and the result will be the annuity required.

Note. — When the amount of an annuity, the time and rate, are given, the annuity may be found by dividing the given amount by the amount of $1 for the given time and rate.

2. The present value of an annuity, to be continued 10 years, at 6 per cent. compound interest, payable annually, is $ 3680.04; required the annuity.

Ans. $ 500. 3. An annuity, remaining unpaid for 9 years, at 5 per cent. compound interest, amounted to $882.125 ; what was the annuity?

4. A yearly pension which has been forborne for 6 years, at 6 per cent., amounts to $ 279; what was the pension ?

Ans. $ 40.

PERMUTATIONS AND COMBINATIONS.

579. PERMUTATION is the process of finding the number of changes that can be made in the arrangement of any given number of things.

580. COMBINATION shows how often a less number of things combined can be taken out of a greater, without respect to their order.

581. To find the number of changes that can be made with any given number of things, taken all at once.

Ex. 1. How many changes of order do the first three letters of the alphabet admit of?

Ans. 6.

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