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COMPOUND INTEREST.

374. COMPOUND INTEREST is interest on the original principal with its interest added when remaining unpaid after becoming due.

When the interest is added to the principal at the end of

every year, and a new principal is thus formed yearly, it is said to compound annually ; when the interest is added to the principal so as to form a new principal half-yearly, it is said to compound semiannually.

375. Compound interest is based upon the principle, that, if the borrower does not pay the interest as it becomes due at stated times, it is no more than just for him to pay interest for the use of it, so long as he shall have it in his possession.

NOTE. — Compound interest is not favored by the laws, though it is not usurious. A contract or promise to pay money with compound interest cannot generally be enforced, being only valid for the principal and legal interest.

376. To find the compound interest of any sum of money at any rate per cent. for any time.

Ex. 1. What is the compound interest of $ 300 for 3 years ?

FIRST OPERATION.

Principal for 1st year,

$ 3 0 0 Interest of $ 1 for 1 year,

.0 6 We first multiply the Interest for 1st year,

given principal by the 18.00

number denoting the

3 0 0.0 0 interest of $1 for one Principal for 2d year,

3 18.00

year, and add the in

terest thus found to .0 6

the principal for the Interest for 2d year,

1 9.0 800 amount; on which as a 31 8.00

new principal we find

the interest for the secPrincipal for 30 year, 33 7.08 ond year, and proceed as

.06 before; and so also with

the third year. From Interest for 3d year,

2 0.2 2 48

the amount of the last 3 3 7 0 8

we subtract the

first principal, and obAmount for 3 years,

3 5 7.3 0 48

tain the compound inFirst principal, 300.00

terest for 3 years Comp. int. for 3 years, $ 5 7.3 0 48, Ans.

year

SECOND OPERATIOX

Principal for 1st year,

$ 3 0 0.0 0 to of the principal,

1 5.0 0 } of the interest at 5 per cent.,

3.0 0 Principal for 2d year,

318.0 0 Amount for 1st year at zo of the principal,

1 5.9 0

[6 per cent. { of the interest at 5 per cent.,

3.1 8 Principal for 30 year,

3 3 7.0 8 Amount for 2d year at to of the principal,

1 6.8 4 5

[6 per cent. of the interest at 5 per cent.,

3.3 70

3 5 7.3 04 Amount for 3d year First principal,

3 0 0.0 0

[at 6 per cent. Compound interest for 4 years, $ 57.3 0 4 Ans.

In the second operation the work is somewhat abridged, by finding the interest for each year at 6 per cent, by taking zo of the principal for the interest at 5 per cent., and of that for interest at 1 per cent.

Rule. - Find the interest of the given sum to the time the interest becomes due, and add it to the principal. Then, find the interest on this amount as a new principal, and add the interest to it, as before. Proceed in the same manner for each successive period when the interest becomes due until the time of settlement.

Subtract the principal from the last amount, and the remainder will be the compound interest.

NOTE. — When partial payments have been made on notes at compound interest, it is customary to find the amount of the given principal, and from it to subtract the sum of the several amounts of the indorsements.

EXAMPLES 2. What is the amount of $ 500 for 3 years at compound interest?

Ans. $ 595.508. 3. What is the compound interest of $ 970 for 2 years 9 months and 24 days?

Ans. $ 173.295. 4. What is the compound interest of $300 for 4 years 6 months, at 7

per
cent. ?

Ans. $ 107.001. 5. What is the compound interest of $ 316 for 3 years 4 months and 18 days?

Ans. $ 69.017. 377. The computation of compound interest is rendered more expeditious by means of the following

TABLE,

bedOWING THE AMOUNT OF ONE DOLLAR AT COMPOUND INTEREST FOR ANY

NUMBER OF YEARS NOT EXCEEDING FIFTY.

No. 3 per cent. 34 per cent. 4 per cent. 5 per cent. 6 per cent. 7 per cent. 1 1.030 000 1.035 000 1.040 000 1.050 000 1.060 000 1.070 000

1.060 900 1.071 225 1.081 600 1.102 500 1.123 600 1.144 900

1.092 727 1.108 718 1.124 864 1.157 625 1.191 016 1.225 043 4 1.125 509 1.147 523 1.169 859 1.215 506 1.262 477 1.310 796 5 1.159 274 1.187 686 1.216 653 1.276 282 1.338 226 1.402 552 6 1.194 052 1.229 255 1.265 319 1.340 096 1.418 519 1.500 730 7 1.229 874 1.272 279 1.315 932 1.407 100 1.503 630 1.605 781 8 1.266 770 1.316 809 1.368 569 1.477 455 1.593 848 1.718 186 9 1.304 773 1.362 897 1.423 312 1.551 328 1.689 479 1.838 459 10 1.343 916 1.410 599 1.480 244 1.628 895 1 790 818 1 407 151 11 1.384 234 1.459 970 1.539 454 1.710 339 1.898 299 2.104 852 12 1 425 761 1.511 069 1.601 032 1.795 856 2.012 196 2.252 192 13 1.468 534

1.563 956 1.665 074 1.885 649 2.132 928 2.409 845 14 1.512 590 1.618 694 1.731 676 1.979 932 2.260 904 2.578 534 15 1 557 967 1.675 349 1.800 944 2.078 928 2 396 558 2.759 032 16

1.604 706 1.733 986 1.872 981 2.182 875 2.540 352 2.952 164 17 1.652 848 1.794 675 1.947 901 2.292 018 2.692 773 3.158 815 18 1.702 433 1.857 489 2.025 $17 2.406 619 2.854 339 3.379 932 19 1.753 506 1.922 501 2.106 849

2.526 950

3.025 600 3.616 526 20 1.806 111 1.989 789 2.191 123 2.653 298 3.207 135 3.869 654 21 1.860 295 2.059 431 2.278 768 2.7 85 963 3.399 564 4.140 562 22 1.916 103 2.131 512 2.369 919 2.925 261 3.603 537 4.430 402 23

1.973 587 2.206 114 2.464 716 3.071 524 3.819 750 4.740 530 24 2.032 794 2.283 328 2.563 304 3.225 100 4.048 935 5.072 367 25 2.093 778 2.363 245 2.665 836 3.386 355 4.291 871 5.427 433 26 2.156 591 2.445 959 2.772 470 3.555 673 4.549 883 5.807 353 27 2.221 289 2.531 567 2.883 369 3.733 456 4.822 346 6.213 868 28 2.287 928 2.620 177 2.998 703 3.920 129 5.111 687 6.648 838 29 2.356 566 2.711 878 3.118 651 4.116 136 5.418 388 7.114 257 30 2.427 262 2.806 794 3.243 398 4.321 942 5.743 491 7.612 255 31 2.500 080 2.905 031 3.373 133 4.538 039 6.088 101 8.145 113 32 2.575 083 3.006 708 3.508 059 4.764 941 6.453 387 8.715 271 33 2.652 335 3.111 942 3.648 381 5.003 189 6.840 590 9.325 340 34 2.731 905 3.220 860 3.794 316 5.253 348 7.251 025 9.978 114 35 2.813 862 3.333 590 3.946 089 5.516 015 7.686 087 10.676 581 36 2.890 278 3.450 266 4.103 933 5.791 816 8.147 252 11.423 942 37 2.985 227 3.571 025 4.268 090 6.081 407 8.636 087 12 223 618 38 3.074 783 3.696 011 4.438 813 6.385 477 9.154 252 13.079 271 39 3.167 027 3.825 372 4.616 366 6.704 751 9.703 507 13.994 820 40 3.262 038 3.959 260 4.801 021 7.039 989 10.285 718 14 974 458 41 3.359 899 4.097 834 4.993 061 7.391 988 10.902 861 | 16.022 670 42 3.460 696 4.241 258 5.192 784 7.761 588 11.557 033 17.144 257 43 3.564 517 4.389 702 5.400 495 8.149 667 12.250 455 18.344 355 44 3.671 452 4.543 342 5.616 515 8.557 150 12.985 482 19.628 460 45 3.781 596 4.702 358 5.841 176 8.985 008 13.764 611 21.002 452

3.895 044 4.866 941 6.074 823 9.434 258 14.590 487 22.472 623 47 4.011 895 5.037 284 6.317 816 9.905 971 15.465 917 24.045 707 48 4.132 252 5.213 589 6.570 528 10.401 270 16.393 872 25.728 907

4.256 219 5.396 065 6.833 349 10.921 333 17.377 504 27.529 930 50 4.383 906 5.584 927 7 106 683 11.467 400 18.420 154 29.457 025

NOTE. — If each of the numbers in the table be diminished by 1, the re. mainder will denote the interest of $ 1, instead of its amount.

Ex. 1. What is the compound interest of $ 360 for 5 years 6 months and 24 days?

Ans. $ 138.14.

OPERATION

Amount of $ 1 for 5 years,
Principal,

$ 1.3 382 2 6

360

80 2 9 3 5 60 401 4 67 80

Amount of $ 360 for 5 years,

481.7 61 360 Amount of $ 1 for 6mo. 24d.,

1.0 34 19 2 7 0 4 5 440 144 5 2 8 4080

481761360 Amount of $ 360 for 5y. 6mo. 24d., 49 8.1 412 462 40 Principal,

3 60. Comp. int. of $360 for 5y. 6mo. 24d., $138.1 4 Ans.

We find the amount of $ 1 for 5 years in the table, and, multiplying it and the number denoting the given principal together, obtain the amount of the $ 360 for 5 years. On this amount as a new principal we find the amount for the remaining 6 months and 24 days, by multiplying by the number denoting the amount of $ 1 for the same time. From the last amount subtracting the original principal, we have left the compound interest required. Hence,

Nsultiply the amount of $1 for the given time and rate, as found in the table, by the number denoting the given principal. The product will be the required amount, from which subtract the given principal, and the remainder will the COMPOUND INTEREST.

Note. — When the given time includes not only the regular periods at which interest becomes due, but also a partial period, as a succession of periods of a year each, followed by one containing months or days, or both, after finding the amount for the regular periods, multiply that amount by the amount of $1 for the remaining time or partial period, and the product will be the required amount for the given time. In like manner, when the number of successive periods exceeds the limits of the table, make the computations for a convenient length of time by means of the table, and on the amount thus found make another computation by means of the table, and so on.

In making computation for a succession of periods shorter or longer than one year each, use the numbers in the table the same as if the periods were those of one year each.

EXAMPLES.

2. What is the compound interest of $ 1200 for 11 years at 7 per bent. ?

Ans. $ 1325.822.

per cent. ?

per cent. ?

3. What is the compound interest of $ 300 for 10 years 7 months and 15 days ?

Ans. $ 257.401. 4. What is the compound interest of $ 5 for 50 years at 7

Ans. $ 142.285. 5. What is the amount of $ 480 for 40 years, at compound interest?

Ans. $ 4937.144. 6. What is the compound interest of $ 40 for 4 years, at 7

Ans. $ 12.431. 7. What is the compound interest of $ 100 for 100 years?

Ans. $ 33830.20. 8. What is the difference between the simple and the compound interest of $ 1000 for 33 years and 4 months ?

9. To what sum will $ 50, deposited in a savings bank, amount, at compound interest for 21 years, at 3 per cent., payable semiannually?

Ans. $ 173.034. (10.) $ 100.

Boston, September 25, 1853. For value received, I promise to pay J. D. Forster, or order, on demand, one hundred dollars, with interest, after six months.

ALLEN T. DAWES.

On this note are the following indorsements: June 11, 1854, received fifty dollars; September 25, 1854, received fifty dollars.

What was due, reckoning at compound interest, August 25, 1855 ?

Ans. $ 2.247. (11.) $ 1000.

St. Paul, January 1, 1850. For value received, I promise to pay Stephen Howe, or bearer, on demand, one thousand dollars, with interest at 7 per cent.

WILSON GOODHUE. Indorsements:- June 10, 1850, seventy dollars; September 25, 1851, eighty dollars; July 4, 1852, one hundred dollars; November 11, 1853, thirty dollars; June 5, 1854, fifty dollars.

At 7 per cent. compound interest, what remains due April 1, 1855 ?

Ans. $ 1022.34.

378. To find the PRINCIPAL, the compound interest, the time, and the rate being given.

Ex. 1. What principal at 6 per cent. compound interest will produce $ 2370 in 10 years ?

Ans. $ 3000.

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