Sometimes when the unit figure is 1, the following form of multiplication may be used with advantage. Example 5. To multiply 7462 by 21. 7462 by 21 14924 156702 In which, putting the multiplier by the side, we let the multiplicand stand for once, and take the product by 2 for 20 times. When the figure of tens is 1, as in the first example of these contractions, the multiplication may be made by the units, and placed one figure to the right. Example 6. To multiply 6375 by 17. 6375 44625 for 7 times 108375 In this form, the first line is considered as a number of tens, and represents 10 times. In some cases of multiplication by numbers between 100 and 112, or between 1000 and 1012, the same principle may be acted upon, by extending the first products 2 or 3 places to the right, according as the first line is intended to stand for 100 or 1000 times. Example 7. To multiply 673 by 108, 112, 1004, and 1012. 673.. by 108 5384 8 times 72684 673.. by 112 8076 12 times 75376 Ex. 57. Multiply 4272 by 21, 41, and 61. 58. Multiply 3067 by 13, 17, and 19. 59. Multiply 2142 by 104 and 112. 60. Multiply 3343 by 1005 and 1012. Ex. 57. 89712-175152-260692. Ex. 59. 222768- 239904. PRODUCTS. 58. 3971-52139- 58273. 39871 60. 3359715-3383116. SIMPLE DIVISION. Simple Division is finding the value or amount of a part of a simple quantity. It is the reverse of Multiplication, for its object is to find a quantity, the product of which multiplied by a given number, will either be equal to a given simple quantity, or be the greatest product that can be obtained out of it. The Dividend is the quantity to be divided. The Divisor is the number by which the division is made, and shows the part which the quotient is to be of the dividend. The Quotient is the product sought by the division, and is always of the same denomination as the dividend. The Remainder is the excess of the dividend above the product of the multiplication of the quotient by the divisor,* when the dividend is greater than the product of any simple quantity multiplied by the given divisor. Rule. Find how often the divisor is contained in the same number (or in one more if it is not contained in the same number) of the highest figures of the dividend; multiply the divisor by this number, subtract the product, and to the remainder bring down the next higher figure of the dividend; then find again how many times the divisor is contained in this increased remainder, and so continue the division until all the figures of the dividend are used; observing, that when the bringing down of one figure from the dividend is not sufficient to make the remainder equal to the divisor, one, two, or more extra figures are to be brought down, but then as many ciphers as extra figures must be placed in the quotient. Observe, also, that when the quotient figure is right, its product will not be greater than the number from which it is to be taken, and the remainder will not be greater than the divisor.-When there are ciphers in the divisor, cut off both these ciphers and the same number of the right hand figures of the dividend. The best Proof of division is multiplication, multiplying the quotient by the divisor, and adding in the remainder, if any. The Remainder may be considered, either as a deficiency of the product of the quotient and the divisor in comparison with the dividend, or it may be valued as a fraction and annexed to the quotient, by making the remainder a numerator and the divisor a denominator. Thus in Ex. 1. the remainder 3 divided by 4 gives 3-4ths expressed by; or if the quantity divided is not in the lowest denomination of its kind, the remainder may be reduced and then divided; thus if the 6375 are pounds of money, then the 3 over are 3 pounds or 60 shillings, which divided by 4 give 15 s.; and thus it makes up the complete quotient £1593 15 s. Otherwise if the 3 are shillings or 36 pence they will give 9 d. or if the 3 are pence, then they are equal to 12 farthings, and will give 3 farthings. When the divisor is a number of which the products are given in the Multiplication Table, it is called short divisor, because the division can then be performed in one line; if the divisor is a composite divisor, that is, one which is exactly the product of two short divisors, it is called a double divisor; and if the divisor is not a short divisor nor exactly a double divisor, it is called a long divisor. Ex. 1. To divide 6375 by 4 4) 6375 4 in 23 gives 5 and 3 over 4 in 15 gives 3 and 3 over Proof. Multiply 1593 by 4 and add in 3. Ex. 2. To divide 16734 by 32 and by 37. 32-4) 16734 8) 4183-2 522-30 1593-3 over 37) 16734 (452 148 193 185 84 74 10 remainder. Observe. In double division if there is a remainder to the second division, we must multiply that remainder by the first divisor, and take in the first remainder; thus, in dividing by 4 we have 2 over, and in dividing by 8 we have 7 over, therefore we multiply 7 by 4=28 and add in 2=30. 1st Proof 522 × 32+30=16734 2nd Proof 452 × 37 +10=16734 Ex. 3. To divide 56740 by 5300 53,00) 567,40 ( 10 53 3740 remainder. There being 2 ciphers in the divisor, we cut off the 40 or the two right hand figures in the dividend, 53 in 56 goes once and 3 over; bring down 7; 53 will not go in 37, set down 0 in the quotient, and to the 37 over bring down the 40 cut off. Ex. 4. Divide 5367 by 100 Quotient 53-67 over. To divide by 10, by 100, or by 1000, cut off 1, 2, or 3 of the lower figures of the dividend; the other figure or figures will be the quotient, and the figure or figures cut off will be the remainder. CONTRACTED METHOD OF LONG DIVISION. The operation of Long Division may be contracted by the omission of the partial products, in the following manner. Rule. When the product of the figure of the quotient multiplied by each figure of the divisor, is not greater than the corresponding figure of the dividend, subtract it and put down the remainder; but when the product is greater, add to the dividend figure as many tens as are necessary to admit of the subtraction, and then putting down only the remainder, carry the number of the tens employed to the next product. Beginning with 58 we say 7 from 8 leaves 1-3 from 5 leaves 2, bring down 4-7 times 5 are 35, from 44 leaves 9 and carry 4-3 times 5 are 15 and 4 are 19, from 21 leaves 2, bring down 2-7 times 7 are 49, from 52 leaves 3 and carry 5-3 times 7 are 21 and 5 are 26, from 29 leaves 3, bring down 0-7 times 8 are 56, from 60 leaves 4 and carry 6-3 times 8 are 24 and 6 are 30, from 33 leaves 3, &c. For other contractions in the operations of Long Division, see the end of Compound Division, and the end of Decimal Division. EXERCISES TO BE PROVED. Ex. 1. Divide 61268 by 29 Ex. 5. Divide 41735 by 101 2. Divide 44842 by 47 3. Divide 18067 by 53 4. Divide 24809 by 79 6. Divide 35387 by 343 7. Divide 91075 by 373 8. Divide 63733 by €75 |