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IV. Common Division may be performed more concisely, by omitting the several products, and setting down only the remainders ; namely, multiply the divisor by the quotient figures as before, and, without setting down the product, subtract each figure of it from the dividend, as it is produced ; always remembering to carry as many to the next figure as were borrowed before.

1. Divide 3104679 by 833.
833) 3104679 (3797545


2. Divide 79165238 by 238. Ans. 332627,12
3. Divide 29137062 by 5317. Ans. 54793314
4. Divide 62015735 by 7803.


Ans. 79474763



Reduction is the changing of numbers from one name or denomination to another, without altering their value.This is chiefly concerned in reducing money, weights, and measures.

When the numbers are to be reduced from a higher name to a lower, it is called Reduction Descending; but when, contrarywise, from a lower name to a higher, it is Reduction Ascending

Before proceeding to the rules and questions of Reduction, it will be proper to set down the usual Tables of money, weights, and measures, which are as follow :


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10 Mills (m)

1 Cent Standard Weight. dwt gr 10 Cents 1 Dime d | The Cent weighs 6 23 Copper 10 Dimes 1 Dollar D Dollar

17 11 Silver 10 Dollars = 1 Eagle E


11 4 Gold The standard for Federal Money of Gold and Silver is 11 parts fine, and 1 part alloy.

A Dollar is equal to 4s and 8d in South Carolina, to 6s in the New-England States and Virginia, to 75 and 6d in NewJersey, Pennsylvania, Delaware, and Maryland, and to 8s in New-York, and North-Carolina.



dwt gr

dit gr 19 B?

9 167

The full weight and value of the English gold and silver coin, is as here be. low : GOLD. Value. Weight SILVER. Value. Weight L $

$.d A Guinea 1 1 0 5 91 A Crown

5 0 Half guinea

0 10 6 2 163 Half-Crown 2 6 Seven Shillings0 7 0 1 194 || Shilling 1 0 3 21 Quarter-guinca 0 5 3

1 81 || Sixpence 06 1 22 The usual value of gold is nearly 41 an ounce, or 2d a grain ; and that of silver is nearly 5s an ounce. Also the value of any quantity of gold, is to the value of the same weight of standard silver, nearly as 15 to 1, or more nearly as 15 and 1-14th to 1.

Pure gold, free from mixture with other metals, usually called fine gold is of so pure a nature, that it will endure the fire without wasting, though it be kept




marked gr

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dwt 24 Grains make i Pennyweight dwt 243 20 Pennyweights 1 Ounce

480= 20= 1 lb 12 Ounces 1 Pound

16 5760=240=12=1 By this weight are weighed Gold, Silver, and Jewels.


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This is the same as Troy weight;.only having some different divisions. Apothecaries make use of this weight in compounding their Medicines; but they buy and sell their Drugs by Avoirdapois weight


continually melted. But silver, not having the purity of gold, will not endure the fire like it; yet fine silver will waste but a very little by being in the fire any moderate time; whereas copper, tin, lead, &c. will not only waste, but may be calcined, or burnt to a powder.

Both gold and silver, in their purity, are so very soft and flexible (like new lead, &c), that they are not so useful, either in coin or otherwise (except to beat into leaf gold or silver), as when they are allayed, or mixed and hardened with copper or brass. And though most nations differ, more or less, in the quantity of such allay, as well as in the same place at different times, yet in England the standard for gold and silver coin has been for a long time as follows-viz. . That 22 parts of fine gold, and 2 parts of copper, being melted together, shall be esteemed the true standard for gold coin : And that il ounces and 2 pennyweights of fine silver, and 18 pennyweights of copper, being melted together, is estemed the true standard for silver coin, called Sterling silver.

* The original of all weights used in England, was a grain or corn of wheat, gathered out of the middle of the ear, and, being well dried, 32 of them were 10 make one pennyweight, 20 pennyweights one ounce, and 12 ounces one pound.

But Vol. I,




marked dr
16 Drmos make 1 Ounce

16 Ounces
1 Pound

28 Pounds
1 Quarter -

4 Quarters 1 Hundred weight cwt
20 Hundred Weight 1 Ton


16 =
256 =
7168 =
28672 =
573440 =


16 =

1 448

28 = 1792 = 112 = 35840 2240

1 cwt
4 = 1
80 20

ton 1

By this weight are weighed all things of a coarse or drossy nature, as Corn, Bread, Butter, Cheese, Flesh, Grocery Wares, and some Liquids ; also all Metals, except Silver and Gold.

oz dwt

= 0

Note, that 11b Avoirdupois =14 11 15} Troy.

0 18 51

1 31 Hence it appears that the pound Avoirdupois contains 69994 grains, and the pound Troy 5760; the former of wbich augmented by half a grain becomes 7000, and its ratio to the latter is therefore very nearly as 700 to 576, that is, as 175 to 144; consequently 144 pounds Avoirdupois are very nearly equal to 175 pounds Troy: and hence we infer that the ounce Avoirdupois is to the ounce Troy as 176 to 192.


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3 Barley-corns make 1 Inch 12 Inches

1 Foot 3 Feet

1 Yard
6 Feet

1 Fathom
5 Yards and a half 1 Pole or Rod
40 Poles

1 Furlong 8 Furlongs

1 Mile 3 Miles

1 League
691 Miles nearly 1 Degree

Deg oro,

But in latter times, it was thought sufficient to divide the same pennyweight into 24 equal parts, still called grains, being the least weight now in common use; and from thence the rest are computed, as in the Tables above.

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2 Inches and a quarter make 1 Nail

NI 4 Nails

1 Quarter of a Yard Qr 3 Quarters

1 Ell Flemish

EF 4 Quarters

1 Yard

Yd 5 Quarters

1 El English E E 4 Quarters if Inch

1 Ell Scotch


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By this measure, Land, and Husbandmen and Gardeners' work are measured ; also Artificers' work, such as Board, Glass, Pavements, Plastering, Wainscoting Tiling, Flooring, and every dimension of length and breadth only.

When three dimensions are concerned, namely, length, breadth, and depth or thickness, it is called cubic or solid measure, which is used to measure Timber, Stone, &c.

The cubic or solid Foot, which is 12 inches in length and breadth and thickness, contains 1728 cubic or solid inches, and 27 solid feet make one solid yard.


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