3. Divide 73146085 Ans. 7597122894. 5201 Ans. 8049614707 11. Suppose 471 men are formed into ranks of 3 deep, what is the number in each rank? Ans. 157. 12. A party, at the distance of 378 miles from the head quarters, receive orders to join their corps in 18 days: what number of miles must they march each day to obey their orders?. Ans. 21. 13. The annual revenue of a nobleman being 37960l.; how much per day is that equivalent to, there being 365 days in the year? Ans. 1047. CONTRACTIONS IN DIVISION. There are certain contractions in Division, by which the operation in particular cases may be performed in a shorter manner as follows: I. 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. This method serves to avoid a needless repetition of ciphers, which would happen in the common way, And the truth of the principle on Ans. 3201998. 0964 3. Divide 7380964 by 23000. 4. Divide 2304109 by 5800. Ans. 3978. 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. Note. 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. 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 ; this must be added to the second remainder, 6, making 64 to the divisor 8, or to be divided by 8. 6X7+1 43 43 43 and this divided by 8 gives 7X8 56 But 64 7 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 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 we proceed 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: Note.-£ denotes pounds, s shillings, and d denotes pence. denotes i farthing, or one quarter of any thing. denotes a halfpenny, or the half of any thing. denotes 3 farthings, or three quarters of any thing. The full weight and value of the English gold and silver coin, both old and new, are as here below. The usual value of gold is nearly 4 an ounce, or 2d a grain; and that of silver is nearly 5s an ounce. Also the value of any quantity of gold, was to the value of the same weight of standard silver, as 15 to 1, in the old coin; but in the new coin they are as 147 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 continually melted. But silver, not having the purity of gold, will not endure the fire like it yet fine silver will waste but 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 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 alloyed, or mixed and hardened with copper or brass. And though most nations differ, more or less, in the quantity of such alloy, 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 11 ounces and 2 pennyweights of fine silver, and 18 pennyweights of copper, being melted together, be esteemed the true standard for silver coin, called Sterling silver. In the old coin the pound of sterling gold was coined into 424 guineas, of 21 shillings each, of which the pound of sterling silver was divided into 62. The new coin is also of the same quality or degree of |