PROOF.-Multiply the divisor and quotient together, and add the remainder, if there be any, to the product: If the work be right, that sum will be equal to the dividend. DIVISION TABLE. 1 2 3 4 5 6 7 8 9 10 11 12, TO USE THE TABLE 2 4 6 8 10 12 14 16 18 20 22 3 91215 18 21 24 27 30 33 4 16 20 24 28 32 36 40 44 GO - Look for the divisor or 24 number by which you wish to divide in the left hand 36 perpendicular column. Then trace the horizon-j 48tal column in which the divisor stands until you find 25 30 35 40 45 50 55 60 the dividend or number 36 42 4854 60 66 49 56 63 70 77 into which you wish to di72 vide, then trace that col umn to the top and you 84 will find the product or number of times the divi 6472 80 88 96 sor is contained in the div idend If you cannot find 81 90 99108 the exact number into which you wish to divide 100 110 120 in the table, look for the next less one, and the dif 121132 ference between them will be what is over. 144 EXAMPLES. 1. How many times is 3 contained in 175817 ? Divisor. Dividend. Quotient. 3) 1758 17 (58605 15 25 24 18 18 1 7 2 Remainder. Here we first write down the dividend, and making a curve on each side, place the divisor (3) at the left hand. In this example, we see, that 3, the divisor, cannot be contained in 1, the first figure of the dividend; therefore we take two figures, (17) and inquire how often 3 is contained therein, which finding to be 5 times, we place the 5 in the quotient, and multiply the divisor by it, setting the first figure of the multiplication under the 7 in 47. How do you prove your work to be right? the PROOF. 58605 Quotient. X3 Divisor+2 175817 the dividend, &c. We then subtract 15 from 17, and find a remainder of 2, to the right hand of which we bring down the next figure of the dividend, viz. 5; then, we inquire how often the divisor 3 is contained in 25, and finding it to be 8 times, we multiply by 8, and proceed as before, till we bring down the 1, when finding we cannot have the divisor in 1, we place 0 in the quotient, and bring down 7 to the 1, and proceed as at the first. Observe that the remainder 2, is here added in multiplying by 3. Note. When there is no remainder to a division, the quotient is the absolute and perfect answer to the question; but when there is a remainder, it may be observed, that it goes so much towards another time as it approaches the divisor; thus, if the remainder be half the divisor, it will go half of a time more, aud so on; in order, therefore, to complete the quotient, put the last remainder to the end of it, above a line, and the divisor below it, as in Example 2. Hence the origin of vulgar fractions, which will be treated of hereafter. The reason of the proof is plain; for, since the quotient is the number of times the dividend contains the divisor, the product of the quotient and divisor must, evidently, be equal to the dividend. As the quotient and divisor are always multiplied during the operation, a simple method of proof is, by adding the several products and remainder (if any) together as they stand. Thus in the above example 1 prod. 2 prod. 15. 3 prod. 4 prod. Remainder 1.5 48. When there is no remainder, what is the quotient?49. When there is a remainder, what is its nature?50. What reason have you for your method of vroof? CASE I. SHORT DIVISION, or when the divisor does not exceed 12. RULE 1. Find how often the divisor is contained in the first figure, or figures of the dividend, setting it under the dividend, and carrying the remainder, if any, to the next figure, as so many tens. 2. Find how often the divisor is contained in this dividend, and set it down as before, continuing so to do, till all the figures in the dividend are used. Note.-The work in Short Division is done mentally, that is, divided in the mind, and the result only written down; whereas in Long Division the operation is written at large. 1. 231 EXAMPLES. In this example, we write down the dividend, and draw a curve line at the left, and a straight line underneath. We then take the first figure of the dividend 9, and find that 4 is contained in 9, 2 times; we write 2 under the 9, and then, mentally, multiply the divisor by it, and the product is 8, which we subtract from 9 and the remainder is 1; we then take the remainder 1, and the next figure of the dividend 2, and the number is 12; we find 4 is contained in 12, 3 times; we write 3 under the 2 and multiply the divisor by it, and the product is 12, which we subtract from 12 and nothing remains; we then take the next figure of the dividend 4, and find that 4 is contained in 4 once; we write 1 under the 4, and multiply the divisor by it, and the product is 4, which we subtract from 4, and nothing remains. All the figures of the dividend having been divided, we find that 4 is contained in 924, 231 times. 2. 3. Quotient, 35967-1 remainder. CASE II. When there is one cipher or more at the right hand of the divisor. RULE.-It or they must be cut off; also, cut off the same number of figures from the dividend, and then proceed as in case first But the figures which were cut off from the dividend must be placed at the right hand of the remainder. 51. What is Short Division ? -52. How is it performed ?-53. When therỡ “ are ciphers at the right hand of the divisor, how do you proceed? Note. To divide by 10, 100, 1000, &c., cut off as many figures from the right hand of the dividend, as there are ciphers in the divisor; the left hand figures will be the quotient, and the right hand figures cut off will be the remainder. CASE III. When the divisor is such a number as any two or more figures in the table multiplied will make. RULE.-Divide the dividend by one of these figures, and the quotient by the other; the last quotient will be the answer. EXAMPLE. 1. What is the quotient of 196473 divided by 72 ? 54. What is your rule, when the divisor is the product of any two figures in the multiplication table? CASE IV. To divide by fractions, or parts of an unit. RULE. If the numerator, or upper figure, is an unit, multiply the given number by the denominator, or under figure, and the product will be the answer: But if the numerator is more than an unit, multiply the given number by the denominator, and divide the product by the numerator. When the divisor is a whole number and a fraction— Multiply the whole number of the divisor by the denominator of the fractional part, and add the numerator to the product for a new divisor; then, multiply the dividend by the denominator of the fraction for a new dividend lastly, divide the new dividend by the new divisor, and the quotient will be the answer. 1. Divide 693 by 242 2421693 EXAMPLES. We multiply 24, the whole number of the divisor, by 4, the denominator of the fractional part, and add the numerator 3 to the 99)2772(28 Ans. product, and the sum is 99 ;-then we multiply 198 792 the dividend 693 by the denominator of the fraction, and the product is 2772;-lastly, we divide 2772 by 99, and the quotient is 28; consequently 243 are contained in 693, 28 times. Ans. 1159 2. Divide 6375 by 53. 3. In one rod there are 5 yards; how many rods are there in 1760 yards? 4. What is the quotient of 10142, divided by 32 ? Ans. 320. Ans. 2766. 55. How do you divide by fractions ?- -56. How, when the divisor is a whole num* ber and fraction? |