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p Look at the statement within parentheses, and say whether the fractions could not as easily be cancelled before writing them, so that the computation could be readily performed by inspection merely. Thus,
(18X X2)=24 rods, as before.
34. If 18 rods of wall can be built by 12 men in 3 days, in what time can 8 men build 24 rods ?
35. If 24 rods of wall can be built by 8 men in 6 days, how many rods can be built by 12 men in 3 days ?
36. If 6 men build a wall 20 feet long, 6 feet high, and 4 feet thick, in 16 days, in how many days will 24 men build one 200 feet long, 8 feet high, and 6 feet thick? The number of days is modified by the number of men, and by the length, the height, and the thickness of the wall. Cancel the four fractions mentally before writing them, and place them together in the form of a simple fraction, the better to bring them under
By inspection, (16x1:1
=80 days, the compound frac126.96.36.199)
tion being equal to 5.
37. If a wall 200 feet long, 8 feet high, and 6 feet thick, can be built by 24 men in 80 days, how many men will build one 20 feet long, 6 feet high, and 4 feet thick, in 16 days? By cancelling before writing, we have
Here the factors 3.2.5 balance the divisors 3•10, leaving 4 as divisor to the 24=6.
38. If 24 men can build a wall 200 feet long, 8 feet high, and 6 feet thick, in 80 days, in what time will 6 men build one 20 feet long, 6 feet high, and 4 feet thick ?
39. If 6 men can build a wall 20 feet long, 6 feet high, and 4 feet thick, in 16 days, how many men will be necessary to build one 200 feet long, 8 feet high, and 6 feet thick, in 80 days?
40. If 60 bushels of oats serve for 15 horses for 16 days, how long will 24 bushels last 8 horses at the same rate? Cancel whilst writing the fraction, and ascertain the result by inspection.
41. If 24 bushels of oats serve for 8 horses for 12 days, how many bushels will be wanted for 15 horses for 16 days at the same rate ? By inspection, after cancelling.
42. If 8 horses eat 24 bushels of oats in 12 days, how many horses may be fed on 60 bushels for 16 days at the same rate ?
43. If 15 horses require 60 bushels of oats for 16 days, how many horses can be fed on 24 bushels for 12 days at the same rate ?
44. If the interest on $100 for 12 months be $6, what will be the interest of $250 for 8 months ?
45. If the interest of $250 for 8 months be $10, wþat will be the interest of $100 for 12 months ?
46. What principal (or sum lent at interest) will gain $10 in 8 months, if $100 gain $6 in 12 months ?
47. The sum of $250 was put at interest until it had gained $10, at the rate of $6 interest for every $100 for 12 months. How long was the $250 lent ?
48. If $100 gain $6 interest in 12 months, how much will $356 gain in 4 months ?
49. If $6 be the interest of $100 for 12 months, in what time will $356 gain $7.12.
50. What principal will gain $7-12 in 4 months, if $6 be the interest for 12 months for $100 ?
51. What will be the interest of $100 for 12 months, if $356 gain $7.12 in 4 months ?
52. What will be the interest of $450 for 24 days, if $100 gain $6 in one year ?
In calculating interest for days, it is customary to consider the year as 360 days, and the months as 30 days each, unless the months are designated.
53. If the interest of $450 for 24 days be $1.80, what will be the interest of $100 for one year ?
54. Nine merchants associated to build a steamboat, for which they advanced equal sums of money. After a while, one of the partners purchased the shares of 6 of the others ;
but afterwards, being pressed for money, sold to a friend onefifth of his entire right. What share of the boat did he retain ? what share did he sell ? and what would be the dividend for the last purchaser, if the boat cleared $45,000. Ans. $7000.
Determinate Fractions, or Compound Numbers. The different methods of increasing and decreasing integers as well as decimal and common fractions, have now, it is believed, been sufficiently exemplified. A class of numbers, however, remains to be noticed, called by some writers DETERMINATE FRACTIONS, from the circumstance of being limited in number and variety of expression, while other fractions are unlimited in both. But, by most arithmeticians, this class is called, rather inappropriately, COMPOUND NUMBERS. These relate chiefly to the division and subdivision of weights and measures, coins, and time. The total want of uniformity in these divisions makes them exceedingly complex and perplex, ing. Old habits, unconnected in their origin, have introduced such a variety, that TABLES of these subdivisions have become absolutely necessary. Strictly speaking, these tables are definitions. They will be found below; and, a knowledge of them being essential to the business of life, they should be thoroughly committed to memory.
In France, during the first revolution, a system of weights and measures was established on the decimal scale, which was, of course, exceedingly simple and intelligible ; and the government of Great Britain is at present engaged in a similar undertaking. The coins of the United States have also been arranged on this scale. But we have not derived all the advantages we might from this simple system, owing to the tenacity with which the people have clung to their old habits of reckoning by pounds, shillings, and pence, — denominations sufficiently perplexing anywhere, but particularly in the United States, as they are not exactly represented by the coins, and as the same denominations possess a different value in the different states. This inconvenience would probably have disappeared long ago, but for the foreign coins which mingle in our circulation. Many attempts have been made in Congress to simplify the system of weights and measures, but hitherto without effect. Such an enterprise, indeed, does not properly belong to an individual nation. To be effectual and thorough, it should be executed by a commission representing all the commercial powers. The movement now making in Great Britain, it is to be hoped, will be followed up by a general congress of scientific men, for the establishment of a system coëxtensive with the field of trade. Meanwhile our youth inust be content to waste their time and burden their memory with these unconnected and unphilosophical divisions of the unit of length and capacity.
TABLES OF COIN, WEIGHT, AND MEASURE.
1. Federal Montey.
1 dime. d.
1 dollar. $. 10 dollars
1 eagle. e.
2. English or Sterling Money.
1 shilling. s.
1 pound. £.
960=240=20=1. Farthings (fourthings) are often written as fractions of a penny.
Thus, 1 farthing is written ^ ; 2 as į; and 3 as .
3. Provincial Currencies. While the United States were British colonies, their cur
rency, like that of the mother country, was sterling. Each colony issued its own money in bills of the denomination of pounds, shillings, and pence. During the revolutionary war, these bills depreciated, and in different degrees in the different colonies, so that a pound or shilling no longer had the same value throughout the land. The federal currency of dollars and cents was adopted soon after the peace. But the people still cling to their old habits of expressing prices in pounds, shillings, and pence; and, as the value of these still differs in different places, it is proper that the student should understand the method of changing a sum of money from one currency into another. This is best done by means of the dollar, which serves as a universal measure, being the uniform standard of value.
The relative value of the dollar, and of the pound, and its subdivisions in the Provincial currencies, is as follows:
a. In English, or Sterling Money. £1=20s=240d.
$1=4s. 6d.=54d.; therefore,
b. In Canada, Nova Scotia, and New Brunswick. £1=20s.
£1=$20=4 $1 ==
c. In New England, Kentucky, and Tennessee. £1=20s.
$1=6s.; therefore, £1=£=42 $1=£6=
d. In New York, North Carolina, and, except Vermont, all the
States added to the Union since 1786. £1=20s.
$1=8s.; therefore, £1=$2l= $1=£=