OPERATION OPERATION. We first inquire how many Divisor 4 ) 9 4 8 Dividend. times 4, the divisor, is contained in 9, the first left-hand figure of 2 3 7 Quotient. the dividend, which is hundreds, and find it contained 2 times, and 1 hundred remaining. We write the 2 directly under 9, its dividend, for the hundreds' figure of the quotient. To 4, the next figure of the dividend, which is tens, we regard as prefixed the 1 hundred that was remaining, which equals 10 tens, and thus form 14 tens, in which we find the divisor 4 to be contained 3 times, and 2 tens remaining. We write the 3 for the tens' figure in the quotient, and the 2 tens that were remaining, equal 20 units, we regard as prefixed to 8, the last figure of the dividend, which is units, in which the divisor 4 is contained 7 times. Writing the 7 for the units' figure of the quotient, we have 237 as the entire quotient, equal the number of yards of cloth at 4 dollars a yard that can be bought for 948 dollars. 2. How many times does 3979 contain 17 ? Ans. 23414 times. We say, 17 in 39, Dividend. 2 times. The 2 we Divisor 17 ) 3 9 7 9 ( 2 3 4 14 Quotient. write in the quo34 tient. 17 x 2 34, which we write 57 under the 39. 39 51 - 34= 5, to which bringing down the 69 next figure of the 68 dividend, we form 57. 17 in 57, 3 1 Remainder. times. The 3 we write in the quotient. 17 X 3 = 51, which we write under the 57. 57 - 51 - 6, to which bringing down the next figure of the dividend, we form 69. 17 in 69, 4 times. The 4 we write in the quotient. 17 X 4 = 68, which we write under the 69. 69 — 68 1, a remainder, or a part of the dividend left undivided. 1 divided by 17 = (Art. 68). The { we write in the quotient, and obtain as the answer required 23417. In this illustration, to render the explanation the more concise, the naming of the denominations of the figures has been omitted. When, as in the operation preceding the last, results only are written down, the method is called short division; and when, as in the last operation, the work is written out at length, it is called long division. The principle is the same in both cases. Hence the general RULE. — Beginning at the left, find how many times the divisor is contained in the fewest figures of the dividend that will contain it, for the first quotient figure. Multiply the divisor by this quotient figure, and subtract the product from the figures of the dividend used. With the remainder, if any, unite the next figure of the dividend. Find how many times the divisor is contained in the number thus formed, and write the figure denoting the result at the right of the former quotient figure. Thus proceed until all the figures of the dividend are divided NOTE 1. — The proper remainder is in all cases less than the divisor. If, in the course of the operation, it is at any time found to be as large as, or larger than, the divisor, it will show that there is an error in the work, and that the quotient figure should be increased. NOTE 2. — If at any time the divisor, multiplied by the quotient figure, produces a product larger than the part of the dividend used, it shows that the quotient figure is too large, and must be diminished. NOTE 3. — It will often happen that, when a figure of the dividend is taken, the number will not contain the divisor; and, in that case, a cipher must be placed in the quotient, and another figure of the dividend taken, and so on, until the number is large enough to contain the divisor. NOTE 4. — If there be a remainder after dividing the last figure of the dividend, write it with the divisor underneath, with a line between them, at the right of the quotient. 74. First Method of Proof. — Multiply the divisor by the quotient, and to the product add the remainder, if any, and if the work be right, the sum thus obtained will be equal to the dividend. NOTE. This method follows from division being the reverse of multiplication. (Art. 72.) 75. Second Method of Proof. — Find the excess of nines in the divisor, quotient, and remainder. Multiply the excess of nines in the divisor and quotient together, and to the product add the excess of nines in the remainder. If the excess of nines in this sum equal the excess of nines in the dividend, the work may be supposed to be right. 76. Third Method of Proof.- Add together the remainder, if any, and all the products that have been produced by multiplying the divisor by the several quotient figures, and the result will be like the dividend, if the work be right. 77. Fourth Method of Proof. — Subtract the remainder, if any, from the dividend, and divide the difference by the quotient. The result will be like the original divisor, if the work be right. NOTE. - The first method of proof (Art. 74) is usually most convenient, and is most commonly employed. OPERATION OPERATION. We first inquire how many Divisor 4 ) 9 48 Dividend. times 4, the divisor, is contained 2 3 7 Quotient. in 9, the first left-hand figure of the dividend, which is hundreds, and find it contained 2 times, and 1 hundred remaining. We write the 2. directly under 9, its dividend, for the hundreds’ figure of the quotient. To 4, the next figure of the dividend, which is tens, we regard as prefixed the 1 hundred that was remaining, which equals 10 tens, and thus form 14 tens, in which we find the divisor 4 to be contained 3 times, and 2 tens remaining. We write the 3 for the tens' figure in the quotient, and the 2 tens that were remaining, equal 20 units, we regard as prefixed to 8, the last figure of the dividend, which is units, in which the divisor 4 is contained 7 times. Writing the 7 for the units' figure of the quotient, we have 237 as the entire quotient, equal the number of yards of cloth at 4 dollars a yard that can be bought for 948 dollars. 2. How many times does 3979 contain 17 ? Ans. 23414 times. We say, 17 in 39, Dividend. 2 times. The 2 we Divisor 17) 3 9 7 9 ( 2 3 4 14 Quotient. write in the quo34 tient. 17 x 2 34, which we write 57 under the 39. 39 51 34= 5, to which bringing down the 69 next figure of the 68 dividend, we form 57. 17 57, 3 1 Remainder. times. The 3 we write in the quotient. 17 X 3 = 51, which we write under the 57. 57 51 = 6, to which bringing down the next figure of the dividend, we form 69. 17 in 69, 4 times. The 4 we write in the quotient. 17 X 4 68, which we write under the 69. 69 - 68 1, a remainder, or a part of the dividend left undivided. 1 divided by 17 - (Art. 68). The i we write in the quotient, and obtain as the answer required 23117. In this illustration, to render the explanation the more concise, the naming of the denominations of the figures has been omitted. When, as in the operation preceding the last, results only are written down, the method is called short division; and when, as in the last operation, the work is written out at length, it is called long division. The principle is the same in both cases. Hence the general RULE. — Beginning at the left, find how many times the divisor is contained in the fewest figures of the dividend that will contain it, for the first quotient figure. Multiply the divisor by this quotient figure, and subtract the product from the figures of the dividend used. With the remainder, if any, unite the next figure of the dividend. Find how many times the divisor is contained in the number thus formed, and write the figure denoting the result at the right of the .former quotient figure. Thus proceed until all the figures of the dividend are divided NOTE 1. — The proper remainder is in all cases less than the divisor. If, in the course of the operation, it is at any time found to be as large as, or larger than, the divisor, it will show that there is an error in the work, and that the quotient figure should be increased. NOTE 2. — If at any time the divisor, multiplied by the quotient figure, produces a product larger than the part of the dividend used, it shows that the quotient figure is too large, and must be diminished. Note 3. — It will often happen that, when a figure of the dividend is taken, the number will not contain the divisor; and, in that case, a cipher must be placed in the quotient, and another figure of the dividend taken, and so on, until the number is large enough to contain the divisor. NOTE 4. — If there be a remainder after dividing the last figure of the dividend, write it with the divisor underneath, with a line between them, at the right of the quotient. 74. First Method of Proof. — Multiply the divisor by the quotient, and to the product add the remainder, if any, and if the work be right, the sum thus obtained will be equal to the dividend. NOTE. This method follows from division being the reverse of multiplica. tion. (Art. 72.) 75. Second Method of Proof. — Find the excess of nines in the divisor, quotient, and remainder. Multiply the excess of nines in the divisor and quotient together, and to the product add the excess of nines in the remainder. If the excess of nines in this sum equal the excess of nines in the dividend, the work may be supposed to be right. 76. Third Method of Proof.- Add together the remainder, if any, and all the products that have been produced by multiplying the divisor by the several quotient figures, and the result will be like the dividend, if the work be right. 77. Fourth Method of Proof. — Subtract the remainder, if any, from the dividend, and divide the difference by the quotient. The result will be like the original divisor, if the work be right. NOTE. The first method of proof (Art. 74) is usually most convenient, and is most commonly employed. OPERATION. PROOF BY THE NINES. = Dividend. Divisor 3 excess. Divisor 759) 18988 (25 Quotient. Quotient 7 excess. 1518 Remainder 4 excess. 3808 37 9 5 (3 x 7) + 4 7 excess. 13 Remainder Dividend 7 excess. 5. Divide 147856 by 97. Ans. 15243 4. OPERATION. PROOF BY ADDITION. Products. Dividend. +97 97 485 2 3 5 194 +194 388 416 28 +388 147 8 5 6 +28 Remainder. 6. Divide 84645 by 285. Remainder Dividend. Ans. 297. OPERATION. PROOF BY DIVISION. Dividend. Dividend. Divisor 2 8 5 8 4 6 45 (2 9 7 Quotient 297) 8 46 45 (2 8 5 Divisor 5 70 594 |