CONTRACTIONS IN MULTIPLICATION. I. When there are Ciphers in the Factors. Is the ciphers be at the right-hand of the numbers; multiply the other figures only, and annex as many ciphers to the right-hand of the whole product, as are in both the factors. When the ciphers are in the middle parts of the multiplier ; neglect them as before, only taking care to place the first figure of every line of products exactly under the figure multiplying with. EXAMPLES 1. 2. Mult 9001635 Mult. 390720400 by - 70100 by 406000 9001635 630. 1445 23443224 15628816 631014613500 Products 158632482400000 3. Multiply 81503600 by 7030. Ans. 572970308000. 4. Multiply 9030100 by 2100. Ans. 18963210000. 5. Multiply 8057069 by 70050. Ans. 564397683450. II. When the multiplier is the Product of two or more Num. bers in the Table ; then * Multiply by each of those parts separately, instead of the whole number at once. EXAMPLES 51307298 7 359151086 8 2873208688 * The reuson of this rule is obvious enough ; for any number multiplied by the component parts of another, must give the same product as if it were multiplied by that number at once. Thus, in the 1st example, 7 times the product of 8 by he given number, makes 56 time: the same number, as plainly as 7 times 8 makes 56, VOL. I. D 2. Mul 2. Multiply 31704592 by 36. Ans. 1141365312. 3. Multiply 29753804 by 72. Ans. 2142273888. 4. Multiply 7128368 by 96. Ans. 684323328. 5. Multiply 160430800 by 108. Ans. 17326526400. 6. Multiply 61835720 by 1320. Ans, 81623150400. 7. There was an army composed of :04 * battalions, each consisting of 500 men ; what was the number of men contained in the whole ? Ans. 52000. 8. A convoy of ammunition † bread, consisting of 250 waggons, and each waggon containing 320 loaves, having been intercepted and taken by the enemy; what is the number of loaves lost? Ans. 80000. OF DIVISION. Division is a kind of compendious method of Subtraction, teaching to find how often one number is contained in another, or may be taken out of it : which is the same thing. The number to be divided is called the Dividend. The number to divide by, is the Divisor.–And the number of times the dividend contains 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, Dividend 12 Divisor 4) 12 (3 Quotient; 4 subtr. showing that the number 4 is 3 times contained in 12, or may be 3 times 8 subtracted out of it, as in the margin. 4 subtr. | Rule-Having placed the divisor before the dividend, as above directed, find how often the divisor is con 4 subtr. tained in as many figures of the dividend as are just necessary, and place the number on the right in the quotient. Mul * A battalion is a body of foot, consisting of 500, or 600,0r 700 men, more or less. + The ammunition bread, is that which is provided for, and distributed to, the soldiers; the usual allowance being a loaf of 6 pounds to every soldier, once in 4 days $ In this way the dividend is resolved into parts, and by trial is found 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 broug:it 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. found how often the divisor is contained in each of those parts, one after another, arranging the several figures of the quotient one after another, into one number. When there is no remainder to a division, the quotient is the whole and perfect answer to the question. 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 ime more ; if the 4th part of the divisor, it will go one fourth of a time mcre; 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 divisor, 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 MethodAdd 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 divi. dend when the work is right. by 4. 3. Divide 73146085 Ads 182865211. 4. Divide 5317986027 by 7. Ans. 7597122897 5. Divide 570196382 by 12. Ans. 47516365 r. 6. Divide 74638105 by 37, Ans, 2017246794 7. Divide 137896254 by 97. Ans. 142161087 8 Divide 3582 1649 by 764. Ans. 4688674 9. Divide 7209 1365 by 5201. Ans. 13861 304 10. Divide 4637064283 by 57606. Ans. 80496 1707. 11. Suppose 471 men are formed into ranks of three 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 the 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 gentleman being 383302 ; how much per day is that equivalent to, there being 365 days in the year? Ans. 1041, 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: 1. Divis 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) 37 419,6 71,00) 310869,01 (4378418 284 Quot. 185209 268 213 556 497 599 568 31 3. Divide • This method is only to avoid a needless repetition of ciphers which would happen in the common way. And the truth of the principle |