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II. When the Multiplier is the Product of two or more Numbers i
* Multiply by each of those parts separately, instead of the whole nu
1. Multiply 51307298 by 56, or by 7 times 8.
2. Multiply 31704592 by 36.
7. There was an army composed of 104 † battalions, each consistin men; what was the number of men contained in the whole?
8. A convoy of ammunition ‡ bread, consisting of 250 waggons, waggon containing 320 loaves, having been intercepted and taken by the what is the number of loaves lost?
DIVISION is a kind of compendious method of Subtraction, teachin how often one number is contained in another, or may be taken out of is the same thing.
The reason of this rule is obvious enough; for any number multiplied by the compon another, must give the same product as if it were multiplied by that number at once. Th example, 7 times the product of 8 by the given number, makes 56 times the same number as 7 times 8 makes 56.
A battalion is a body of foot, consisting of 500, or 600, or 700 men, more or less. The ammunition bread is that which is provided for, and distributed to the soldier allowance being a loaf of 6 pounds to every soldier, once in 4 days.
The number to be divided, is called the Dividend.—The number to divide by, is the Divisor. And the number of times the dividend con.ains the divisor, is called the Quotient.-Sometimes there is a Remainder left, after the division is finished.
The usual manner of placing the terms, is, the dividend in the middle, having the divisor on the left-hand, and the quotient on the right, each separated by a curve line; as to divide 12 by 4, the quotient is 3,
showing that the number 4 is 3 times contained in 12, or may be three times subtracted out of it, as in the margin.
* Rule. Having placed the divisor before the dividend, as above directed, find how often the divisor is contained in as many figures of the dividend as are just necessary, and place the number on the right in the quotient. Multiply the divisor by this number, and set the product under the figures of the dividend before-mentioned.-Subtract this product from that part of the dividend under which it stands, and bring down the next figure of the dividend, or more if necessary, to join on the right of the remainder.-Divide this number, so increased, in the same manner as before; and so on till all the figures are brought down and used.
N. B. If it be necessary to bring down more figures than one to any remainder, in order to make it as large as the divisor, or larger, a cipher must be set in the quotient for every figure so brought down more than one.
TO PROVE DIVISION.
+ MULTIPLY the quotient by the divisor; to this product add the remainder, if there be any; then the sum will be equal to the dividend when the work is right.
⚫ In this way we resolve the dividend into parts, and find by trial how often the divisor is contained in each of those parts, one after another, and arranging the several figures of the quotient one after another, into one rumber.
When there is no remainder to a division the quotient is the whole and perfect answer to the ques tion. But when there is a remainder, it goes so much towards another time as it approaches to the divisor: so, if the remainder be half the divisor, it will go the half of a time more; if the 4th part o the divisor, it will go one-fourth of a time more; and so on. Therefore, to complete the quotient, set the remainder at the end of it, above a small line, and the divisor below it, thus forming a fractional part of the whole quotient.
+ This method of proof is plain enough: for since the quotient is the number of times the dividend contains the divi-or, the quotient multiplied by the divisor must evidently be equal to the dividend.
There are also several other methods sometimes used for proving Division, some of the most useful of which are as follow:
Second Method -Subtract the remainder from the dividend, and divide what is left by the quotient; so shall the new quotient from this last division be equal to the former divisor, when the work is right.
Third Method. Add together the remainder and all the products of the several quotient figures by the divisor, according to the order in which they stand in the work; and the sum will be equal to the dividend when the work is right.
11. Suppose 471 men are formed into ranks of 3 deep, what is the nun each rank?
12. A party, at the distance of 378 miles from the head quarters, orders to join their corps in 18 days; what number of miles must they each day to obey their orders?
13. The annual revenue of a gentleman being 37960/; how much a day equivalent to, there being 365 days in the year? An
CONTRACTIONS IN DIVISION.
THERE are certain contractions in Division, by which the operation ticular cases may be performed in a shorter manner; as follows:
I. Division by any Small Number, not greater than 12, may be exped performed, by multiplying and subtracting mentally, omitting to set do work, except only the quotient immediately below the dividend.
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 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 t, and the whole will be the true remainder; otherwise, the figures cut off only will be the remainder.
1. Divide 3704196 by 20.
Quot. 185209 18
3. Divide 7380964 by 23000. 4. Divide 2304109 by 5800.
2. Divide 31086901 by 7100.
554407-6 second rem. Ans. 55440713.
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:
1. Divide 31046835 by 56, or 7 times 8. 7) 31046835
8) 4435262-1 first rem.
making 6; to the divisor 8, or to be divided by 8. But 6; =
6 the last rem.
mult. 7 preced. divisor.
This method is only to avoid a needless repetition of ciphers, which would happen in the common way. And the truth of the principle upon 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 2d 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-1 he 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, 6×7+1 43 -- ; and this divided by 8 7 7
8. Divide 7014596 by 72.
3. Divide 5130652 by 132. 4. Divide 83016572 by 240.
IV. Common Division may be performed more concisely, by omitting several products, and setting down only the remainders; namely, multiply divisor by the quotient figures as before, and, without setting down the pro subtract each figure of it from the dividend, as it is produced; always re bering to carry as many to the next figure as were borrowed before.
REDUCTION is the changing of numbers from one name or denomina another, without altering their value. This is chiefly concerned in re money, weights, and measures.
When the numbers are to be reduced from a higher name to a lowe called Reduction Descending; but when, contrariwise, from a lower na higher, it is Reduction Ascending.
Before proceeding to the rules and questions of Reduction, it will be to set down the usual Tables of money, weights, and measures, which below:
OF MONEY, WEIGHTS, AND MEASURES.
*£ denotes pounds, a shillings and d denotes pence.
denotes 1 farthing, or one quarter of any thing.
The full weight and value of the English gold and silver coin, is as here bel
Value. £ 8. d.
A Guinea 1 1 0
Weight. dwt. gr. 5 94 2 164
1 0 0