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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.

2d power.

3d power.

∞∞4th power.

Ratio 3,

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 1st 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 døl. per bu. amount to 14187 dols. Ans.

2. Sold 24 yards of Holland, at 2d. for the first yard, 4d. for the second, 8d. for the third, &c. how much did it amount to? Ans. 139810L. 2s. 6d.

3. Bought 30 bushels of wheat, at 2d. for the first bushel, 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.

S8947818 L. 10s. 6d. Amount.

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?

Received 214469929 L. 5s. 3d. Ans. {Gained 214466929 L. 5s. 3 d.

5 A goldsmith sold 1 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 4 L. per ounce for the gold?

He received 5825 L. 8s. 51d.

Ans. And gained 5777 L. 8s. 54d.

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. 5s. 34d.

and

7. A person married his daughter on new year's day, 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. 7s. 6d.

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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.

[blocks in formation]

Thus, 103 1.007417; 1.03 — 1.014889; and 1.007417 X 1.014889 1.022416, for 3 quarters.

Rate

per cent.

TABLE I.

Amounts of L. for a year and for Simp. int.
quarters, at Compound Interest.

Ratio. For 3qrs.[For 2qrs. For 1qr.

of L. 1 for
1 month.

3 1.03 1.022416 1.014889 1.007417 .002500
31 1.035 1.026137 1.017349 1.008637 .002917
1.04 1.0298521.019804 1.009853-003333
4층 1.045 1.033563 1.022252 1.011065 .003750
1.05 1.037270 1.0246951.012272 .004167
1.055 1.0409731.027132 1.013475 .004583
1.06 1.044671 1.029536 1.014674 .005000
61 1.065 1.048364 1.031988 1.015868 .005417
1.07 1.052053 1.034408 1.017058 .005833

5층

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 for 3, &c. thus, 1.06 x 1.06 x 1.06 × 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 × 1.044671 = 1.318873 for 43 years, &c.

The power or the amount of 1L. or dol. may also be ob tained for months and days (nearly) by adding the monthly simple interest of 1 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 yr. 1.029563: For 42 yrs. = 1.318873 Int. of 1L. for 1 mo. = .005000 One sixth for 5 dys. = .000833

.005000 .000833

For 7 mo. 5 dys. =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.

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

yr 31 per ct. 4 per ct. 41 per ct. 5 per ct. 51 per ct. 6 per ct.

11.0350000 1.0400000 1.0450000 1.0500000 1.0550000 1,0600000 21.0712250 1.0816000 1.0920250 1.1025000 1.1130250 1.136000 31.1087178 1.1248640 1.1411661 1.1576250 1.1742413 1.1910160 41.1475230 1.1698585 1.1925186 1.2155062 1.2388246 1.2624769 51.1876863 1.2166529 1 2461819 1.2762815 1.3069598 1.3382256 61.2292553 1.2653190 1.3022601 1.3400956 1.3788426 1.4185191 7 1.2722792 1.3159317 1.3608618 1.4071004 1.4546789 1.5036302 81.3168090 1.3685690 1.4221006 1.4774554 1.5346862 1.5938480 91.3628973 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 1.5394540 1.6228530 1.7103393 1.8020919 1.8982985 12 1.51106861.6010322 1.6958814 1.7958563 1.9012069 2.0121964 13 1.5630560 1.6650735 1.7721961 1.8856491 2.0057732 2.1329282 14 1.6186945 1.7315764 1.8519449 1.9799316 2.1160907 2.2609039 415 1.6753488 1.8009435 1.9352824 2.0789281 2.2324756 2.3965581 16 1.7339860 1.8729812 2.0223701 2.1828745 2.3552617 2.5403517] 17 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 2.6078603 2.5269502 2.7656458 3.0255995 20 1.9897888 2.1911231 2.4117140 2.6532977, 2.9177563 3.2071355 212.0594314 2.2787680 2.5202411 2.7859625 3.0782329 3.3995636 22 2.1315115 2.3699187 2.6336520 2.9252607 3.2475357 3.6035374 23 2.2061144 2.4647155 2.7521663 3.0715237 3.4261502 3.8097496 24 2.2833284 2.5633041 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 3.5556726 4.0231279 4.5493829 27 2 5315671 2.8833685 3.2820095 3.7334563 4.2443999 4.8223459 28 2.6201719 2.9987033 3.4296999 3.9201291 4.4778419 5.1116867 29 2.7118779 3.1186514 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 3.91385744-5380394 5.2580671 6.0881007 32 3.0067075 3.5080587 4.0899810 4.7649414 5.5472608 6.4533867 33 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 4.6673478 5.5160152 6.5138230 7.6860868 36 3.4502661 4.1030325 4.8773784 5.7918101 6.8720832 8.1472520 37 3.5710254 4.2680898 5.0968604 6.0814069 7.2500478 8.6360871 383.6960113 4.4388134 5.3262192 6.3854772 7.6488004 9.1542523 39 3.8253717 4.6163659 5.5658990 6.7047511 8.0694844 9.7035074 40 3.9592597 4.8010206 5.8163645 7.0399887 8.5133060 10.2857178 414.0978337 4.9930614 6.0781009 7.3919881 8.9815378 10.9028608 42 4.2412579 5.1927838 6.3514246 7.7615871 9.4755224 11.5570325 43 4.3897020 5.4004952 6.6375522 8.1496669 9.9966761 12.2504547 44 4.5433415 5.6165150 6.9362421 8.5571502 10.5464933 12.9854817 45 4.7023585 5.8411756 7.2483730 8.9850077 11.1265504 13.7646107 464.8669411/6.0748236 7.574549719.4342581 11.7385217|14.5904873]

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