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Note. The first term in this question is 4, the ratio 3, the number of terms 20 : therefore raise the ratio to the 19th power, which is one less than the number of terms.

Ist power.
L2d power.

hun 3d power.
op og 4th power.

Ratio 3, 9, 27, 81

81

81 648

6561 8th power of the ratio.
6561

6561 39366 32805 39366

43046721 16th power.

27 3d power.

301327047
86093442

1162261467 19th power.

4 Ist term.

4649045868

3 Ratio.

13947137604

4 First term.

Ratio less one 2) 13947137600

7680)6973568800 Sum of the series.

908016 pints, = 14187 bushels. 14187 bushels, at 1 dol. per bu. amount to 14187 dols. Ans.

{lainenced 214469929 L. SS. 34

2. Sold 24 yards of Holland, at ?d, for the first yard, 4d. for the second, sd. for the third, &c. how much did it amount to?

Ans. 139810L. 2s. 60. 3. Bought 30 bushels of wheat, at 2d. for the first bush. el, 4d. for the second, 8d. for the third, &c. what does the whole amount to, and what is the price per bushel on an average ?

Ans.

58947818 L. 10s. 6d. Amount.

298261 L. 12s. 4d. per bushel. 4. A merchant sold 30 yards of lace at 2 pins for the first yard, 6 for the second, 18 for the third, &c. and disposed of the pins at 1000 for a farthing? how much did he receive for the lace ? and how much did he gain by the sale, supposing the lace cost him 100L. per yard?

Ans.
ş

55. d. 5. A goldsmith sold i lb. of gold at a farthing for the first ounce, a penny for the second, 4d. for the third, &c. in quadruple proportion; how much did he receive for the whole, and how much did he gain by the sale, supposing he gave 4L. per ounce for the gold?

He received 5825 L. Ss. 5 d. Ans.

And gained 5777 L. 85. 5 d. 6. What sum would purchase a horse with 4 shoes, and eight nails in each shoe, at one farthing for the first nail, a halfpenny for the second, 'a penny for the third, &c. doubling to the last.

Ans. 4473924L. 55. 3 d. 7. A person married his daughter on new year's day, and gave her one dollar towards her portion, promising to double it on the first day of every month for one year; what was her portion?

Ans. 4095 dols. 8. Suppose a man wrought 20 days, and received for the first day's labour 4 grains of corn, for the second 12, for the third 36, &c. what did he receive for his labour, supposing 7680 grains to make a pint, and the whole to be sold at 2s. 6d. per bushel ? Ans. 1773L. 78. Odi

{.

COMPOUND INTEREST,

BY DECIMALS.

The ratio in compound interest is the amount of one pound or dollar for one year; which is thus found : As 100 : 105 :: 1: 1.05. As 100 : 105.5 ::1:1.055,

For quarterly amounts, take the 4th root of the ratio ; for half yearly, the square root; and, for 3 quarters, the product of the quarterly and half yearly. 4

2 Thus, ✓ 1.03 = 1.007417; v1.03. - 1.014889; and 1.007417 X 1.014889 = 1.022416, for 3 quarters.

TABLE I.

Amounts of L. I for a year and for
Rate
quarters, at Compound Interest. of L. 1 for

Simp. int.
per
cent.

1 month.
Ratio.For 3qrs./For2qrs. For Iqr.

3
3

1.03 1.022416|1.014889/1.007417 .002500
1.035 1.0261371.017349 1.008637 .002917
1.04 1.029852 1.019804 1.009853,003333
1.045 1.0335631.022252 1.011065 .003750
1.05 1.0372701.024695 1.012272 .004167
1.055 1.040973/1.027132|1.013475 .004583
1.06 1.04467 11.0295361.014674 .005000
1.065 1.048364 1.031988 1.015868 .005417
1.07 11.0520531.03440811.017058 .005833

COMPOUND INTEREST BY DECIMALS. 177 The ratio involved to the time is the amount of 1 L. or dol. for the time given; as a square for 2 years, a cube sor 3, &c. thus, 1.06 X 1.06 X 1.06 X 1.06.

4

or 1.06 = 1.262477 = the 4th power of 1.06, or the ratio involved to 4 years. When the ratio is to be involved to years and

quarters, the power for the years is to be multiplied by the proper quarterly amount; as, 1.262477 x 1.044671 1.318873 for 4 years, &c.

The power or the amount of i L. or dol. may also be obtained for months and days (nearly) by adding the monthly simple interest of i L. or dollar, or proper parts thereof, to the amount of the quarter next preceding the expiration of the given time, for what that time exceeds the said quarter; thus,

Amount for i yr. = 1.029563: For 4 yrs. = 1.318873 Int. of IL. for I mo. = .005000

.005000 One sixth for 5 dys. = .000833

.000833

For 7 mo. 5 dy8. = 1.035396: For 4y. 10mo. 5d, = 1.324706.

The ratio may be thus involved to any time whatever; but the operation is facilitated by the following tables ; which may be extended to 100 years, or upwards, by multiplying the amount for 46, by that for the time above 46, &c.

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TABLE II. Shewing the amount of L. 1. or dol. from 1 year to 46.

|yr|31 per ct. 4 per ct. 42 per ct. 5 per ct. 5 per et.

per ct.

6 1 1.03500001.0400000/1.0450000 1.0500000 1.0550000 1,0600000 21.0712250 1.0816000 1.0920250 1.1025000 1.1130250 1.136000 31.10871781.1248640 1.1411661 1.1576250 1.1742413 1.1910160 4 1.14752301.1698585 1.19251861.2155062 1.2388246 1.2624769 5 1.18768631.21665291 2461819|1.2762815 1.3069598 1.3382256 61.2292553 1.2653190 1.3022601 1.3400956 1.3788426 1.4185191 71.2722792/1.3159317 1.36086181.4071004 1.4546789 1.5036302 - 81.3168090 1.3685690 1.42210061.4774554 1.5346862 1.5938480

9 1.3628973 1.4233118 1.4860951|1.5513282 1.6190939 1.6894789 10 1.4105987|1.4802442 1.5529694/1.6288946 1.7081440 1.7908476 11 1.4599697 1.53945401.6228530 1.7103393 1.8020919 1.8982985 121.5110686 1.60103221.69588141.7958563 1.9012069 2.0121964) 13/1.5630560 1.6650735/1.7721961 1.8856491 2.0057732 2.1329282 114 1.618694.5/1.7315764 1.85194491.9799316 2.1160907 2.2609039

15 1.6753488 1.80094351.9352824 2.0789281 2.2324756 2.3965581 116 1.7339860|1.8729812 2.0223701 2.1828745 2.3552617| 2.54035171 117 1.7946755 1.9479005 2.1133768 2.2920183 2.4848011 2.6927727 18/1.8574892 2.0258161 2.2084787 2.4066192 2.6214652 2.8543391 19 1.9225013 2.1068491 2.80786032.5269502 2.7656458 3.0255995 201.9897888|2.19112312.41171402.6532977). 2.9177563 3.2071355 1212.0594314 2.2787680 2.5202411 2.7859625 3.0782329 3.3995636

222.13151152.3699187 2.63365202.9252607 3.2475357 3.6035374 123 2.2061144 2.4647155 2.7521663 3.0715237 3.4261502 3.8097496 24 2.28332842.5633041 2.8760138|3:2250999 3.6145885 4.0489346 25 2.3632449/2.6658363|3.0054344 3.3863549 3.8133910 4.2918707) 26 2.4459985 2.7724697 3.1406790|3.55567264.0231279 4.5493829 27 2 5315671 2.8833685 3.2820695 3.7334563 4.2443999 4.8223459 128 2.6201719 2.9987033|3.42969993.9201291 4.4778419 5.1116867 29 2.71187793.1186514/3.5840364 4.1161356 4.7241232 5.4183870) 30 2.8067937 3.2433975 3.7453181 4.3219423 4.9839469 5.7434912 31 2.9050314|3.3731334|3.9138574 4.5380394 5.2580671 6.0881007 132 3.0067075 3.5080587 4.08998104.7649414 5.5472608 6.4533867) 35 3.1119423 3.6481831 4.2740301 5.0031885 5,8523600 6.8405899

34 3.2208603|3.7943163 4.4663015|5.2533479 6.1742398 7.2510253 35 3.3335904 3.9460889 4.66734785.5160152 6.5138230 7.6860868 3613.4502661 4.1030325 4.8773784 5.7918101 6.8720832 8.1472520 137|3.5710254|4.2680898 5,09686046.0814069 7.2500478 8.6360871 38 3.6960113 4.4388134 5.32621926.3854772 7.6488004 9.1542523 39 3.8253717 4.6163659 5.5658990 6.7047511 8.0694844 9.7035074 140 3.9592597 4.8010206 5.81636457.0399887 8.5133060|10.2857178 414.0978337 4.99306146.07810097.3919881| 8.981537810.9028608 42 4.2412579 5.19278386.35142407.7615871 9.4755224 11.5570325 43 4.3897020 5.4004952/6.6375522 8.1496669 9.9966761 12.2504547 44 4.5433415 5.61651506.9362421 8.557150210.546493312.9854817 45 4.7023585 5.84117567.2483734 3.9850077 11.1265504 13.7646107 4614.866941116.074823617.574549719.4342581|11.7385217|14.5904873

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