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5. REDUCTION OF DECIMALS.
Case I. To reduce Vulgar Fractions to Decimals. RULE.-Annex a cipher to the numerator, and divide it by the denominator; annex a cipher to the remainder, and divide as before, and so on; the quotient will be the decimal required.*
Examples. 1. Reduce to a decimal. 3. Reduce } to a decimal. 4)3.0.75 Ans.
Ans. .333 + 28
4. Reduce 1 to a decimal.
Aus. ,5 20 20
5. Reduce il to a decimal.
Ans. 7045+ 2. Reduce à and to decimals. 6. Reduce 4 to a decimal. Ans. .5, .25
Ans. .175 +
Case II. To reduce numbers of different denominations to their equivalent decimal values.
Rule*-Write the numbers perpendicularly under each other for diviilends, proceeding orderly from the least to the greatest, and write on the left hand of each, for a divisor, the number which it takes of that to make one of the next higher denomination, and draw a line perpendicularly between the divisors and dividends.
Begin with the highest, and write the quotient of each division as decimal parts, on the right of the next dividend below. Continue thus to do till all the dividends are used, and the last quotient will be the decimal required.
Examples. 1. Reduce 12s. 9d. 3q. to the decimal of a pound.
The given numbers stand as integers. In the first place I 4 3 annex 2 ciphers to 3, making it 3.00, and dividing it by 4, the 12 9.75 quotient .75, the decimal of a penny, which I set against the 20 | 12.8125 9d. then dividing by 12, I get .8125; and then by 20, I get
.640625, the decimal of a pound. .640625 dec. required.
* If the number of figures in the quotient be just qual to the number of ciphers annexed to the dividend, then the quotient is the true decimal; but if it be less, it must be made equal by placing ciphers at the left hand. In cases where the numerator is greater than the denominator, it is an improper fraction, and the quotient will be a whole or mixed number.
† The reason of this rule will appear by a little attention to the first example. Here 3 qrs. is of a penny, which, reduced to a decimal, is .75. Hence 9 75 =91d. But 9.75 is 178 of a penny=127 of a shilling, which reduced to a decimal is .8125, and therefore 12. ožd. may be expressed thus, 12 8125 In like manner, 12.8125s. is 6th of a shilling=188148 of a pound, which reduced to a decimal is .640625£ as found by the rule.
2. Reduce 10 oz. 18 pwt. 16 4. Reduce 19s. 54d. to the grs. to the decimal of a pound. decimal of a pound. Ans..911111+
Ans. .971875 3. Reduce 3 qrs. to the deci. 5. Reduce 2 qrs. 21 lb. 12 oz.. mal of a shilling Ans. .0625. 10 dr. to the decimal of a cwt.
avoirdupoist Ans. .6945452
Case III. To reduce shillings, pence and farthings to the decimal of a pound
by inspection. Rule. *-Write half the greatest even number of shillings for the first decimal figure, consider how many farthings there are in the given pence and farthings, and let these possess the second and third places; renembering to increase the second place by 5, if the shillings be an odd number, and the third place by 1, when the farthings exceed 12, and by 2 when they exceed 36.
Examples. 1. Reduce 13s. 10}d. to the 3. Find the decimal value of decimal of a pound, by inspec91d. by inspection. tion.
Ans: .040 6=1 of 12, the greatest even dum[ber of shillings.
4. Find the decimal value of 5 for the odd shilling.
6s. by inspection.
Ans. .300 42=farthings in 104d. 2 because the farthings exceed 36.
5. Find the decimal value of
1s. 10d. by inspection. .694 the decimal required.
Ans. .092 2. Find the decimal, value of 6. Find the decimal value of 15s. Sud. by inspection.
16s. 4 d. by inspection. Ans. .785
* As shillings are so many 20ths of a pound, half the shillings are so many 10ths of a pound; therefore half the even number of shillings will occupy the place of tenths in the decimal. When there is an odd shilliog, it is just equal
a tenth, or a Ito part of a pound; it is therefore properly expreseed by a 5 in the second decimal place. A pound is equal to 960 farthings ; now had it happened that 1000 qrs. instead of 960, had made a pound, farthings would have been so many thousands of a pound, and might have been placed in the decimal as such. "But 960 falls short of 1000 just 4 part of itself; consequent. ly, any number of farthings, increased by its 24 gart, will be an exact decimal expression for it. When the farthings are over 12, and less than 36, a ak is more than , and less than 14, and therefore 1 must be added to give the nearest decimal in the third place ; and when the farthings exceed 36, a te part is more than 1), and therefore 2 must be added. This gives the decimal sufficiently correct for common practice; but when greater exactness is required, a 4 is to be found by division, and the decimal places increased.
Case IV. To find the value of any given decimal in the terms of an integer. Rule.-Multiply the decimal by that number which it takes of the next less to make one of the denomination in which the decimal is given, and cut off from the right hand as many places for a remainder as there are places in the
given decimal. Proceed with the remainder in the same way, and so on through all the denominations ; the numbers standing on the left of the parts cut off will form the answer.
Examples. 1. What is the value of .640625 2. What is the value of .972916 of a pound?
of a pound ? .640625 The examples up
Ans. 19s. 5d.
iqr. 20 der this case are the
examples under Case 3. What is the value of.911111-4 12.812500 11. inverted, by which of a pound troy ? 12 it will be seen that: Ans. 10oz. 18pwt. 15grs.*
they reciprocally 9.750000 prove each other.
4. What is the value of .0625 4
of a shilling?
Ans. 3qrs. -3.900000 Ans. 12s. 9d. 3qrs.
Case V. * To find the value of any decimal of a pound by inspection. Rulet-Double the first figure in the decimal for shillings, and if the second figure be 5, or more than 5, reckon another shilling ; then call the figures in the second and third places, (after 5, if contained in the second, is deducted,) so many farthings ; abating 1 when they are above 12; and 2, when they are above 36 ; and the result is the answer.
By comparing the answer to this example, with its correspondent sum in Case II. it will appear that a grain is lost. But it will be seen that the decimal which remains, approaches very nearly to another integer, and the loss is because the complete value of the decimal is not employed in the operation. It is usually best to take the lowest denomination to the nearest integer ; that is, when the first figure of the decimal is more than 5, add 1 to the integer. In this way, the numbers in the example will agree with their correspondent numbers in Case II.
If, after having obtained the integers of the lowest denomination, a decimal remain, and it is required to find its value precisely, change it to a Vulgar Fraction, and reduce it to its lowest terms by dividing both parts of the fraction as long as you can find any number which will divide them both without a remainder. Supposing the decimal were .75, it 18 76. Now 75 and 100 may both be divided by 25; thus, 25) 1664, and as there is no number which will divide 3 and 4 without a remainder, is the lowest, or njost simple expression for 70
+ This rule is the converse of that given under Case III. and the reason of it must be sufficiently obvious, from what was there said.
Examples. 1. Find the value of .785 of a 4. Find the value of .040 of a pound by inspection.
pound by inspection. 14s. double 7.
Ans. 9 d. ls. for 5 in the 2d place.
5. Find the value of .092 of a 8d. 3qrs.= 35 qrs. abat. 5 from 8.. 1qr. for exces of 12, abated,
pound by inspection.
Ans. 18. 10d. 158.81.2qrs. Ans.
6. Find the value of .819 of a 2. Find the value of .875 of a pound by inspection. pound by inspection.
Ans. 16s. 4 d. Ans. 17s.. 60..
7. Find the value of .694 of a 3. Find the value of .3 of a pound by inspection. pound by inspection.
Ans. 138. 104d. Ans. 6s..
Application of the preceding Rules in Fractions..
1. What is the sum of 84 rods, 6. Into how many pieces # of a 124 rods, and 23 rods, when ad-foot long, may a pole 15.5 feet ded together?
long, be cut ?
Ans. 62. 8.5
7. What is the value of .875 of 12 25 2.75
a day? Ans. 21 hours.
8. How many feet of boards 23.50 Ans.
will cover the two sides of a The scholar should bear in mind barn, they being each 361 feet that where these expressione, 4, , long, and 14.2 feet high? and & occur, he is to substitute for
Ans. 1036.6 feet. them their equivalent decimal values, which are respectively.25, -5 and.75. 9. How many acres in a piece
2. From 154 rods, take 33 of land 74.8 rods long, and 45.6 rods.
Ans. 11.75 rods. broad? 3. Multiply 124 feet by 34
Ans. 21 acres, 50.88 rods. feet. Ans. 39.8125 feet.
10. How many rods in a piece 4. How many feet in a board of land 16; rods long, and 15% 94 feet long, and 24 feet wide ? wide ? Ans. 21.375 feet.
Ans. 255.75 rods,=1 acre, 5. How much wood'in a pile
95% rods. 20 feet long, 33 feet wide, and 11. How many solid feet in a 64 high?
pile of timber 25 feet long, 14.2. Ans. 437.5 feet=3. cord's, 53 | feet wide and 14.2 feet high? feet, 864 inches,*
Ans. 5041 feet..
* The feet are brought into cords by dividing by 128, the number of feet in a cord. The .5 of a foot is reduced to inches by Case IV. Redaction of Decimals.
QUESTIONS. 1. What is a Decimal Fraction !
pointing off in each of these 2. What would be its denominator rules. were it expressed ?
11. How are Vulgar Fractions re3. What is a whole number and de duced to Decimals ?
cimal in the same sum called ? 12. How are numbers of different 4. How does the value of the num denominations reduced to their bers vary from unity ?
equivalent decimal valu ? 5. What effect have ciphers placed 13. How are shillings, pence, and
on the right and left hand of farthings reduced to the deciDecimals?
mal of a pound by inspection ? 6. What is the rule for the addition 14. How is the value of a given deof decimals ?
cimal found in the terms of an 7. For Subtraction?
integer ? 8. For Multiplication ?
15. How do you find the value of the 9 For Division ?
decimal of a pound by inspec10. Repeat in short the method of tion ?
3 Federal Money.* FEDERAL MONEY is the established coin of the United States. Its denominations are in a decimal or ten-fold proportion, and were determined by act of Congress, August 8, 1786.
The different denominations in Federal Money are exhibited in the following
Table. 10 Mills, m.
Cent, marked ct.
$ or doli.
* For its simplicity and ease of reckoning, Federal Money is superior to any other, and it is fast supplanting, as it should do, the old method of computing by pounds, shillings, pence, and farthings. Could the same improvements be made in weights and measures, a competent knowledge of Arithmetick could be obtained with one half the labor which is now required, and the same computa'ions could be made in half the time.
The standard for gold and je eleven parts fine, and one part alloy, or, as goldsmiths would say, 22 carats fine. A carat is ak part of any quantity, and when gold or silver is said to be 22 carats fine, it is to be understood, that were the whole mass divided into 24 parts, 22 of them would be pure gold or silver, and the other 2 alloy. Copper is commonly used as alloy in gold and silver, and is employed to render them more hard and durable. The weight of our coins are as foilows : E-gle, 11 pwts. 6 grs. Half Eagle, 5 pwts. 15 grs. Dlar, 17 pwts. 7 grs. Haif dollar, 8 pwts. 16 grs. and 100 cents weigh 211b. avoirdupois. The denominations less than a dollar are expressive of their valurs ; thus, mill is froin the Latin mille, a thousand; for 1000 mills make 1 dollar ; cent is from centum, a kundred, because 100 cents make 1 dollar, and dime is from the French signifying the tenth part, because 10 dimes make 1 dollar. Uncined gold, 22 carats fine, is worth at the mint, $209.79 per pound troy, and vocoined silver, of the same fiueness, is worth $9.92 per pound troy.