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ĐIVISION OF INFINITE DECIMALS.
RULE. Change both divisor and dividend into vulgar fractions, find their quotient in a vulgar fraction, and reduce it to a decimal. 39. What is the quotient of .36 by .25? .36==11
* ==ix==1183 Then, 125}=1.4229249011857707509881 Ans. 40. What is the quotient of 234.6 by .7? 41. What is the quotient of 13.5i69533 by 4.2927 42. Divide 319.28007112 by 764.5.
RELATIONS OF NUMBERS.
Any number may be viewed as a part, or as so inany parts of any other number, and it is in this view, that we shall, at present, notice the RELATION of one number to another.
For example, 1 is } of 5, 3 is of 5, 9 is of 5, &c. Here 1 stands in the same relation to 5 that } does to a unit; 3 stands in the saine relation to 5 that does to a unit; and 9 stands in the same relation to 5 that does to a unit. Thus, the number which is viewed as the part or parts of another, becomes a numerator, and the other number the denominator of a vulgar fraction. This fraction may be reduced, and the relation it expresses will remain unaltered. For instance, g of 8 is the same as of 8; and ?of 8 is the same as a
or i of 8. In the various practice of arithmetic, most of the solutions are performed by process to which the performer is led, by considering the relation which exists between the numbers concerned. The truth of this remark will appear evident to the learner, in the course of subsequent exercises
1. Express 16 as a fractional part of 56, and reduce the fraction to its lowest terms.
2. Express 9 as the fractional part of 45, and reduce the fraction to its lowest terms.
3. What part of 34 is 20? What part of 34 is 21?
4. What part of 34 is 49?-Or, in other words, what is the improper fraction that expresses the relation in which 49 stands to 34 ?
5. What part of 24 is 36 ? What part of 24 is 37?
6. What part of 2 yards 1ft. 6 in. is lyd. 2 ft. 10 in.?
In this example, 2yd. I ft. 6 in. becomes a denominator, and lyd. 2ft. 10 in. the numerator. But both these quantities must be reduced to their lowest denomination, inches; the relation will then be simple, and may admit of being reduced to lower terms.
7. What part of 1 yard is 2 feet 6 inches ? 8. What part of £3 14 s. is 16 s. 10d.? 9. What part of 9 s. 7 d. 2 qr. is 2 s. 9d. Iqr.? 10. What part of 5 gallons 2 pints is 3 quarts 3 gills ?
What part of 2 acres is 1 acre 3 roods 32 roda?
12. What part of $7 is $4.65 ?
$7=700 cents, and $4.65=465 cents. Then 465 cents is 463 of 700 cents. 46=1
When either or both the numbers, whose relation is to be expressed, contains a decimal fraction, the decimal places in the two numbers must be made equal — if they are not already so-by annexing decimal ciphers. The decimal points may then be erased, and the numbers written as the terms of a vulgar fraction. For example, the relation of .14 to 9 is 76=iso:
13. What part of 2.1 is 1.72?
21. What part of 5.8 is 31.42?
. What part of .253 is .97 ?
23. What part of is is?
The expression of this relation is, at first, a complex fraction, of which is is the numerator, and the denominator. The expression may be simplified by reducing these fractions to a common denominator, and taking the new numerators for the terms of the relation. to reduce a complex fraction to a simple one, page 43.
24. What part of 12 is 10%?
42. If 35 horses eat 12278 pounds of hay in a week, what will 17 horses eat, in the same time?
The most obvious view of the solution of this question is this — If 35 horses eat 12278 pounds, 1 horse will eat 35 of 12278 pounds, which is 35035 pounds; and 17 horses will eat 17 times 350 pounds, which is 5963}} pounds. A more concise view, however, may be taken, as follows. 17 horses are of 35 horses, and they will eat of the 12278 pounds of hay. Therefore, we shall obtain the answer by multiplying 12278 pounds by the fraction }}. 12278X3=5963} Ans.
43. If a car run 552 miles upon a rail-road, in 24 hours, how far will it run in 13 hours ?
44. If a car run 3 miles [960 rods] in 8 minutes (480 seconds), in what time will it run 300 rods?
45. If a hogshead of wine [63 gallons] cost $98.50, what will 45 gallons cost, at the same rate ?
46. If the annual expense of supporting a fort manned with 600 soldiers be $.182571, what is the expense of a fort manned with 424 soldiers ?
47. If I can buy 325 barrels of four for $ 1425, how many barrels can I buy for $ 521?
43. If a ferry boat cross the river 18 times in 5 hours, in how many hours will it cross 4 times ?
49. If 9 barrels of flour cost $ 32, what will 28 bl. cost?
In this example, the relation in which 28 barrels stand to 9 barrels is expressed by an improper fraction; 28 barrels being of 9 barrels. Therefore the answer is obtained by multiplying $32 by 26; that is, by multiplying $32 by 28, and dividing the product by 9.
50. If it take 300 yards of cloth to make the uniform clothes for 52 soldiers, how many yards are required to clothe 784 soldiers ?
51. If 12 horses eat 20 bushels of oats in a week, how many
bushels will 45 horses eat in the same time? 52. If a post 5 feet high cast a shadow 3 feet, on level ground, what is the height of a steeple, which, at the same time, casts a shadow 176 feet ?
53. If $40 will pay for 14 yards of cloth, how many yards can be bought for $75 ?
54. If 95 bushels of corn cost $ 68.25, what will 320 bushels cost, at the same rate ?
55. Suppose a ship's expenses in Liverpool to be £131 135. 10d. for 22 days; what would be her expenses in the same port for 35 days?
56. If 144 bushels of corn will grow upon 3 acres 1 rood 15 rods of land, how much land is necessary to produce 500 bushels ?
57. Bought 269 yards of cloth, at the rate of $ 100 for 30 yards.
What did it amount to?
58. Bought 24 yd. 3qr. 'I na. of cloth, at the rate of $ 12.30 for 4 yd. I qr. 2 na.
What did it amount to? Since it is necessary, in this example, to consider 24 yd. 3qr. 1 na. as a fractional part of 4 yd. I qr. 2 na., the first step in the operation is, to reduce both quantities of cloth to nails.
59. If 13 gal. 2qt. 1 pt. of wine cost $ 21.15, what will 36 gal. 3 qt. 1 pt. cost, at the same rate ?
60. If 26 barrels of four cost £ 28 14 s. 6d. how many barrels will £ 35 10s. 4 d.
pay for ? 61. If 6 gal. 2qt. 1 pt. of wine will fill 31 bottles, how many bottles are required for 11 gal. 3 qt.?
62. If 144 gross of buttons cost £22 19 s., how many gross can be bought for £12 5s. 51 d.?'
63. If 2 hhd. 19 gal. 2qt. of wine cost £93 1 s. 24d. what will 25 hhd. 36 gal. cost ?
64. If 15 yards of cloth cost $39.45, how many yards can be bought for $21? (See remark under example 12.)
65. At the rate of $ 94 for 78 days' work, in how many days can a labourer earn $ 72.375?
66. At the rate of $ 240 for 9.5 acres of land, what is the value of 7.25 acres ?
67. At the rate of $132.50 for 8 acres of land, what is the value of 12.7 acres ?
68. At the rate of 75 cents for .92 of a bushel of corn, what is the value of .648 of a bushel ?
69. In how many minutes will a locomotive car run 49.9 miles; allowing it to run at the rate of 2.5 miles in 5.75 minutes ?
70. If 43.64 pounds of copper be worth $ 9.075, what is the value of 108.9 pounds ?
71. If 14 dollars will pay for the carriage of a ton 75.6 miles, what distance can a ton be carried for 16 dollars 75 cents, at the same rate ?
72. If of a yard of cloth cost $7, what is the cost of is of a yard ? (Recur to example 23.)
73. If a rail-road car run 260 miles in 12 hours, what distance will it run in 10 hours ? (See example 24.)