NOTE 3.--The pupil will point off the decimal places in the quotient of this and the following example, as directed by the rule. 7. Divide 5737 by 13.3. Quot. 431353. 8. What is the quotient of 2464.8 divided by 008 ? 9. Divide 2 by 53.1. 10. Divide 012 by 005. Ans. 308100. Quot. '037+. Quot. 2.4. " .075. 11. Divide three thousandths by four hundredths. Examples in U. S. Currency. 1. Divide $59.387 equally among 8 men; how much will each man receive? OPERATION. 8)59-387 of Ans. $7.4233, that is, 7 dollars, 42 cents, 3 mills, and another mill. The is the remainder, after the last division, written over the divisor, and expresses such fractional part of another mill. For most purposes of business, it will be sufficiently exact to carry the quotient only to mills, as the parts of a mill are of so little value as to be disregarded. 2. At $75 per bushel, how many bushels of rye can be bought for $141? Ans. 188 bushels. 3. At 12 cents per pound, how many pounds of butter may be bought for $37? Ans. 296 pounds. 4. At 64 cents apiece, how many oranges may be bought for $8? Ans. 128 oranges. 5. If 6 of a barrel of flour cost $5, what is that per barrel? Ans. $8.333+. NOTE 4.-If the sum to be divided contain only dollars, or dollars and cents, it may be reduced to mills, by annexing ciphers before dividing; or, we may first divide, annexing ciphers to the remainder, if there shall be any, till it shall be reduced to mills, and the result will be the same. 6. If I pay $468-75 for 750 pounds of wool, what is the value of 1 pound? Ans. $0.625; or thus, $62. 7. If a piece of cloth, measuring 125 yards, cost $181-25, what is that a yard? Ans. $1.45. 8. If 536 quintals of fish cost $1913-52, how much is that a quintal? Ans. $3.57. 9. Bought a farm, containing 84 acres, for $3213; what did it cost me per acre? Ans. $38.25. 10. At $954 for 3816 yards of flannel, what is that a yard? Ans. $0.25. 11. Bought 72 pounds of raisins for $8; what was that a pound? Ans. $0-111; or, $0.111+. 12. Divide $12 into 200 equal parts; how much is one of the parts? how much? 12 200 13. Divide $30 by 750. 14. Divide $60 by 1200. Ans. $.06. 60 = how much? 1200 15. Divide $215 into 86 equal parts; how much will one of the parts be? 215 how much? 67. Review of Decimal Fractions. Questions.-61. Decimal Fractions, why so called? Divisions and subdivisions of a unit? Decrease and increase, how? How written and operated? How distinguished from integers? Denominator, what about it? To write decimals? To read decimals? Table? 62. How is the value of every decimal determined? Decimals of different denominators, how reduced to a common denominator? Note 2, what advantages over common fractions? How are common fractions reduced to decimals? 63. United States Currency. The Unit, U. S. Currency? Denominations, and their value? Alloy? Standard gold and silver for coinage, U. S. currency? Carat, all about it? Weight of the Eagle, &c., with other questions at the discretion of the Teacher. Note 2? Note 3 64. Addition and Subtraction of Decimals. Ex. 1 and 2, with their solutions? Rule? 65. Multiplication of Decimals. Ex. 1, and solution? Decimal Diagram, its purpose, and all about it? Its application illustrating any operation in the multiplication of decimals? Ex. 2, its solution, and placing the point? Rule for the multiplication of decimals? 66. Division of Decimals. Ex. 1, solution and application of the Decimal Diagram? Ex. 2, solution and placing the point? Note 1? Note 2? Rule for the division of Decimals? EXERCISES. 1. A merchant had several remnants of cloth, measuring as follows, viz.: 7 yards. 761 59/000/004/0 1 81 316 แ How many yards in the whole, and what would the whole come to, at $3.67 per yard? NOTE 1.-Reduce the common fractions to decimals. Do the same wherever they occur in the examples which follow. Ans. 36-475 yards. $133.863+, cost. 2. From a piece of cloth, containing 365 yards, a merchant sold, at one time, 73 yards, and, at another time, 12 yards; how much of the cloth had he left? Ans. 16.7 yards. 3. A farmer bought 7 yards of broadcloth for $3311, two barrels of flour for $146, three casks of lime for $75, and 7 pounds of rice for $3; what was the cost of the whole? 4. At 12 cents per pound, what will 373 pounds of butter cost? Ans. $4.7183. 5. At $17.37 per tun for hay, what will 11 tuns cost? Ans. $201.92§. 6. The above example reversed. At $201.92% for 11g tuns of hay, what is that per tun? Ans. $17.37. 7. If 45 of a tun of hay cost $9, what is that per tun? Ans. $20. 8. At 4 of a dollar a gallon, what will 25 of a gallon of molasses cost? Ans. $1. 9. What will 2300 pounds of hay come to, at 7 mills per pound? Ans. $16.10. 10. What will 765 pounds of coffee come to, at 18 cents per pound? Ans. $137.79. 11. Bought 23 firkins of butter, each containing 42 pounds, for 163 cents a pound; what would that be a firkin? and how much for the whole? Ans. $159.39 for the whole. 12. A man killed a beef, which he sold as follows, viz., the hind quarters, weighing 129 pounds each, for 5 cents a pound; the fore quarters, one weighing 123 pounds, and the other 125 pounds, for 4 cents a pound; the hide and tallow, weighing 163 pounds, for 7 cents a pound; to what did the whole amount? Ans. $35.47. 13. A farmer bought 25 pounds of clover seed at 11 cents a pound, 3 pecks of herds-grass seed for $2.25, a barrel of flour for $6.50, 13 pounds of sugar at 123 cents a pound; for which he paid 3 cheeses, each weighing 27 pounds, at 84 cents a pound, and 5 barrels of cider at $1.25 a barrel. The bal ance between the articles bought and sold is 1 cent; is it for or against the farmer? 14. A man dies, leaving an estate of $71600; there are demands against the estate amounting to $39876-74; the residue is to be divided between 7 sons: what will each one receive? Ans. $4531-8942. . 15. How much coffee, at 25 cents a pound, may be had for 100 bushels of rye, at 87 cents a bushel? Ans. 348 pounds. 16. At 12 cents a pound, what must be paid for 3 boxes of sugar, each containing 126 pounds? Ans. $47.25. what will they all Ans. $56387.50. 18. A merchant sold 275 pounds of iron, at 61 cents a pound, and took his pay in oats, at $0.50 a bushel; how many bushels did he receive? Ans. 34.375 bushels. 17. If 650 men receive $86.75 each, receive? 19. How many yards of cloth, at $4.66 a yard, must be given for 18 barrels of flour, at $9.32 a barrel? Ans. 36 yards. 20. What is the price of three pieces of cloth, the first containing 16 yards, at $3.75 a yard; the second, 21 yards, at $4.50 a yard; and the third, 35 yards, at $5.12a yard? Ans. 333.871. 68. INFINITE DECIMALS. There are two sorts of Decimal Fractions-complete or finite, and incomplete or infinite. The common fraction =4.00÷66 (=·0606+) cannot be reduced exactly, for however long the division be continued there will be the same remainder. The figures 06, which continually repeat, are called Repetends or Circulating Decimals. If one figure repeats, as 333, it is called a single repetend, and is denoted by writing only the circulating figure, 3, with a point over it, signifying that the figure, 3, will be the quotient, however long the division be continued, forming an infinite or never-ending series of 3s. If the figures which repeat be two, as in the above example, 06, or if they be more than two, as in the repetend, 425, they are called compound repetends, and are denoted by two dots, one over the first and the other over the last figure of the repetend. If other figures arise before those which circulate, it is called a mixed repetend; as, 17248+, where 17 is the finite part, and 248 the repetend. Similar repetends are those which begin at the same place; thus, 13 and 72; 264; 9038. Dissimilar repetends are those which begin at different places; thus, 6127; 405; 347. Conterminous repetends are those which end at the sanre place; thus, 749; 506; 684. Similar and conterminous repetends are those which begin and end at the same place; thus, 1308; 4012; 7654. 1 1 NOTE 1.-The denominator of every decimal fraction is 10 or some power of 10, and is produced by the prime factors 2 and 5. It follows, that whenever the denominator is a prime number other than 2 or 5, or when it contains any prime factor except 2 or 5, the common fraction cannot be accurately expressed in decimals. 2, 3, 3, and 1250, decimally expressed, are ·5, ·2, ·125, and ·0008, complete and finite. But 3, 4, and are 3+, 142857+, and 27+, incomplete or infinite. This will still be the fact, though we continue the division forever, constantly approximating to the true value of the common fraction, but never reaching it. This is probably the greatest paradox in the science of Arithmetic, and is analogous to the greatest wonder in Physics, the infinite divisibility of matter. Any quotient obtained by annexing ciphers to the remainder is known to be infinite, whenever a remainder occurs which has occurred before; and every quotient, which does not terminate, must, at some place, repeat or circulate. This truth, rarely understood, inevitably follows from the fact, that in dividing one number by another continually, the greatest number of different remainders can never exceed the number of units in the divisor less one; and for every different remainder only, there can be a different quotient figure. When all the possible remainders have arisen, if not before, the same figures must repeat, and in the same order, in the quotient. Hence the name of circulating or periodical decimals. 69. Reduction of Infinite Decimals. I. To reduce a simple repetend to its equivalent common fraction. If we attempt to reduce to a decimal, we obtain a continued repetend of the figure 1; thus, 111+, that is, the repetend i. The value of the repetend 1, then, is ; the value of 222+, the repetend 2, will evidently be twice as much, that is, 3. In the same manner, ·3=3; ·4=4; ·5=§; |
