Dividend, Divisor 4) 12 (3 Quotient; showing that the number 4 is 3 times contained in 12, or may be 3 times subtracted out of it, as in the margin. 12 4 subtr. 8 4 subtr. 4 4 subtr. *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. Note. 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 the dividend is resolved into parts, and by trial is 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 time more; if the fourth part of 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 divisor, the quotient multiplied by the divisor must evidently be equal to the dividend. There are 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. D M + R Fourth Method, by casting out the nines. Make a cross as in multiplication, and cast out the nines from the divisor and quotient, and place the respective remainders, instead of D and Q respectively. Cast the nines also out of the remainder, and annex it to Q by the sign plus, at R. Multiply D by Q, and add in the number R; and from this also cast out the nines. Place the result at M: and if this last number be the same as that left after casting out the nines from the dividend, the work is probably correct. M 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 379607.; 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 more concisely: 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. Dividend, 12 4 subtr. 8 4 subtr. 4 subtr. * 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. Note. 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. * In this way the dividend is resolved into parts, and by trial is 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 time more; if the fourth part of 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 divisor, the quotient multiplied by the divisor must evidently be equal to the dividend. There are 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. M Q+ R M 1707 Ans. 8049688 10. Divide 4637064283 by 57606. 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 379607.; 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 more concisely 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 Divisors; 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. III. When the Divisor is the exact Product of two or more of the Numbers not greater than 12; divide by one of the factors of the divisor, putting down the remainder to the right of the quotient, but separated by the mark (; then this quotient by the next of the factors, setting down the remainder to the right of the quotient, as in the former case; then this quotient by the next factor, and so on till all the factors have been used. The final quotient is the integer part of the quotient required. * This method serves to avoid a needless repetition of ciphers, which would occur in the common way. And the truth of the 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. |