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1. Division by any Small Number, not greater than 12, may be expeditiously performed, by multiplying and subtracting mentally, omitting to set down the work, except only the quotient immediately below the dividend.
II. * When Ciphers are annexed to the Divisor; cut off those ciphers from it, and cut off the same number of figures from the right-hand of the dividend; then divide with the remaining figures, as usual. And if there be any thing remaining after this division, place the figures cut off from the dividend to the right of it, and the whole will be the true remainder; otherwise, the figures cut off only will be the remainder.
EXAMPLES. 1. Divide 3704196 by 20. 2. Divide 31086901 by 7100. 2,0) 370419,6 71,00) 310869,01 ( 43787135.
284 Quot. 18520915
* This method is only to avoid a needless repetition of ciphers, which would happen in the commop way. And the truth of the
3. Divide 7380964 by 23000,
Ans. 3207986. 4. Divide 2304109 by 5800.
360o III. When the Divisor is the exact Product of two or more of the small Numbers not greater than 12: * Divide by each of those numbers Separately, instead of the whole divisor at once.
N. B. There are commonly several remainders in working by this rule, one to each division; and to find the true or whole remainder, the same as if the division had been performed all at once, proceed as follows; Multiply the last remainder by the preceding divisor, or last but one, and to the product add the preceding remainder ; multiply this sum by the next preceding divisor, and to the product add the next preceding remainder ; and so on, till you have gone backward through all the divisors and remainders to the first, As in the example following:
EXAMPLES. 1. Divide 31046835 by 56 or 7 times 8. 7) 31046835
6 the last rem.
mult, 7 preced, divisor. 8) 4435262-1 first rem.
43 whole rem.
principle on which it is founded, is evident ; for, cutting off the same number of ciphers, or figures, from each, is the same as dividing each of them by 10, or 100, or 1000, &c. according to the number of ciphers cut off; and it is evident, that as often as the whole divisor is contained in the whole dividend, so often must any part of the former be contained in a like part of the latter.
* This follows from the second contraction in Multiplication, being only the converse of it; for the half of the third part of any thing, is evidently the same as the sixth part of the whole; and so of any other numbers. The reason of the method of finding the whole remainder from the several particular ones, will best appear from the nature of Vulgar Fractions. Thus, in the first example above, the first remainder being 1, when the divisor is 7, makes 4 ; this must be added to the second remainder, 6, making 64 to the divisor 8, or to be divided by 8. But 64 = 6x7+]
2. Divide 7014596 by 72.
Ans. 9742491 3. Divide 5130652 by 132.
Ans. 38868 4. Divide 83016572 by 240.
Ans. 345902-92. 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.
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
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 :
* £ denotes pounds, s shillings, and d denotes pence.
denotes i farthing, or one quarter of any thing.
denotes 3 farthings, or three quarters of any thing. The full weight and value of the English gold and silver coin, is as here below : GOLD. Value. Weight, SILVER. Value. Weight, of sd dwt gr
d A Guinea
A Crown 5 19 81 Half-guinea 0 10 6 2 161 Half-crown 2 6 9 164 Seven Shillings 0 70
1 0 3 21 Quarter-guinea 5 3 18 Sixpence Q 611 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 1.5 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
dwt 24 Grains make 1 Pennyweight dwt 24 = 1 20 Pennyweights Ounce
480= 20 = 1 12 Ounces 1 Pound
lb | 5760=240 =12=1 By this weight are weighed God, Silver, and Jewels.
1 480 = 24 =
1 1b 5760 = 288 = 96 = 12 = 1 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 Avoirdupois weight.
without wasting, though it be kept 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 it ounces and 2 pennyweights of fine silver, and 18 pennyweights of copper, being relted together, is esteemed 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 to make one pennyweight, 20 penny