PROOF. 58605 Quotient. X3 Divisor+2 175817 dend, &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 Observe that the remain-multiply by 8, and proceed as before, till der 2, is here added in we bring down the 1, when finding we multiplying by 3. cannot have the divisor in 1, we place 0 in the quotient, and bring down 7 to the 1, and proceed as at first. 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 hall ofa time more, and 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 atand. Thus in the above example 15.. 1 prod. 2 prod 24. 3 prod. 4 prod. 15 Remainder .2 48. When there is no remainder, what is the quotient? -49. When there is a re mainder, what is its nature? -59. What reason have you for your method of proof? 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 to do so, 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. 4)924 EXAMPLES. In this example, we write down the dividend, and draw a curve line at the left, and a straight line under231 neath. We then take the first figure of the dividend 9, and we 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 is 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 in the dividend having been divided, we find that 4 is contained in 924, 231 times. $2. Quotient, 35967-1 remainder. CASE II. 3. 1 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 there 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 a unit. RULE. If the numerator or upper figure, is a 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 a 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. EXAMPLES. 1. Divide 693 by 242 We multiply 24, the whole number of the divisor, by 4, the denominator of the fractional part, and add the numerator 3 to the prod99)2772(28 Ans. uct, and the sum is 99;-we then multiply 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. 198 792 792 2. Divide 6375 by 5 Ans. 1159 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 number and fraction? FEDERAL MONEY. FEDERAL MONEY is the coin of the United States, established by Congress in 1786. The Gold Coins are the Eagle, Half Eagle, and Quarter Eagle. The Silver Coins are the Dollar, Half Dollar, Quarter Dollar, Dime and Half Dime. The Copper Coins are the Cent, and Half Cent. Mill is only imaginary, - there being no Coin less than a half cent. The denominations of Federal Money are Eagles, Dollars, Dimes, Cents and Mills. In keeping accounts, and in reading Federal Money, eagles and dimes are not named; eagles being read tens, &c. of dollars ;and dimes, tens of cents. Dollars are separated from cents by a point or comma;-all the figures at the left hand of the point are dollars the two first at the right hand are cents, and the third is mills. When there is a fourth figure, it represents tenths of mills-the fifth, hundredths of mills, &c. Thus, $25,7525 are read twenty-five dollars, seventy-five cents, two mills, and five tenths of a mill. By an act of Congress it was resolved, that there should be Standard. 270 grs. Two Copper coins; viz. 2. The Half Cent 135 416 208 104 83,2 41,6 20,8 208 ,05 104 104 Any sum in Federal Money may be read either in the lowest denomination, or partly in the higher, and partly in the lowest; thus, $54,321 may be read 54321 mills, or 5432 cents 1 mill, or 543 dimes 2 cents 1 mill, or 5 eagles 4 dollars 3 dimes 2 cents 1 mill; all which denominations may be easily distinguished by the decimal point, thus The method best adapted to practical purposes, and which has been sanctioned by a law of the United States, is the decimal form of expression by a decimal point, making the dollar the money unit. Dollars, therefore, will occupy the place of units, and the less denominations will be decimal parts of a dollar, and distinguished by the decimal point. 57. What is Federal Money?- 58. What are the different denominations ?59. What denominations are named in keeping accounts?- 60. How are dollars separated from cents?61. What are the figures at the left, and at the right of the point? E |