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to a decimal fraction.
The numerator must be reduced to hundredths, by annexing two ciphers, before the division can begin.
66) 4'00 ('0606+, the Answer.
As there can be no tenths, a cipher must be placed in the quotient, in tenth's place.
Note. cannot be reduced exactly; for, however long the division be continued, there will still be a remainder.* It is sufficiently exact for most purposes, if the decimal be extended to three or four places.
From the foregoing examples we may deduce the following general RULE :-To reduce a common to a decimal frac
* Decimal figures, which continually repeat, like '06, in this example, are called Repetends, or Circulating Decimals. If only one figure repeats, as 3333 or 7777, &c, it is called a single repetend. If two or more figures circulate alternately, as 060608, “234234234, &c, it is called a compound repetend. If other figures arise before those which circulate, as 743333, 143010101, &c., the decimal is called a mixed repetend.
A single repetend is denoted by writing only the circulating figure with a point over it: thus, 3, signifies that the 3 is to be continually repeated, forming an infinite or never-ending series of 3's.
A compound repetcnd is denoted by a point over the first and last repeating figure: thus, 234 signifies that 234 is to be continually repeated.
It may not be amiss, here to show how the value of any repetend may be found, or, in other words, how it may be reduced to its equivalent vulgar fraction.
If we attempt to reduce to a decimal, we obtain a continual repetition of the figure 1: thus, '11111, that is, the repetend 'i. The value of the repetend 'i, then, is; the value of will evidently be twice as much, that is, 3. †, and '4 = §, and '5 = §, and so on to 9, 1. What is the value of '8?
222, &c., the repetend 2,
2. What is the value of '6? Ans. §. of? of $4? of '5? If be reduced to a decimal, it produces '010101, or the repetend ċi. The repetend '02, being 2 times as much, must be and '48, being 48 times as much, must be 8, and '74
and '03 — y3,
What is the value of '3?
tion,-Annex one or more ciphers, as may be necessary, to the namerator, and divide it by the denominator If then there ho a remainder, annex another cipher, ana tuae us unjure, and so continue to do so long as there shall continue to be a remainder, or until the fraction shall be reduced to any necessary degree of exactness. The quotient will be the decimal required, which must consist of as many decimal places as there are ciphers annexed to the numerator; and, if there are not so many figures in the quotient, the deficiency must be supplied by prefixing ciphers.
EXAMPLES FOR PRACTICE.
4. Reduce, 1, 4%, and 1129 to decimals.
Ans. '5; 25; '025; '00797+. 6 to decimals.
5. Reduce, Todo, 1795, and Ans. '692+; '003; '0028 +; ‘000183 + 6. Reduce 171, 17, 38ʊ to decimals.
7. Reduce 1,9, 080, 1; 3, T'I, 1, 935 to decimals.
8. Reduce }, }, I, t, t, t, t, zb, 25, 7 to decimals.
If y be reduced to a decimal, it produces 001; consequently, 2002=735, and ‘037 37 99 3, and 425 = 3, &c. As this principle will apply to any number of places, we have this general RULE for reducing a circulating decimal to a vulgar fraction,--Make the given repetend the numerator, and the denominator will be as many 9s as there are repeating figures.
3. What is the vulgar fraction equivalent to 704?
4. What is the value of '003? $2463? - ·002103 ?
Ans. Fot. 01021 ? 33335.
5. What is the value of 43?
In this fraction, the repetend begins in the second place, or place of hundredths. The first figure, 4, is fʊ, and the repetend, 3, is 3 of To, that is,; these two parts must be added together.
5+ 5% = 33
= 43, Ans. Hence, to find the value of a mixed repetend,—Find the value of the two parts, separately, 6. What is the value of 153?
and add them together.
= 18$ = ·73, Ans
7. What is the value of ‘0047? 8. What is the value of '138 ? It is plain, that circulates may be added, subtracted, multiplied, and divided, by first reducing them to their equivalent vulgar fractions
REDUCTION OF DECIMAL FRACTIONS.
¶ 75. Fractions, we have seen, (¶ 63,) like integers, are reduced from low to higher denominations by division, and from high to lower denominations by multiplication.
To reduce a compound num- To reduce the decimal of a ber to a decimal of the highest higher denomination to integers denomination. of lower denominations.
1. Reduce 7 s. 6 d. to the decimal of a pound.
2. Reduce 375 £. to integers of lower denominations. 6 d. reduced to the decimal 375 £. reduced to shillings, of a shiling, that is, divided that is, multiplied by 20, is by 12, is '5 s., which annexed 750 s.; then the fractional to the 7 s. making 7'5 s., and part, '50 s., reduced to pence, divided by 20, is 375 £. the that is, multiplied by 12, is 6 d. Ans. 7 s. 6 d. The process may be pre- That is,-Multiply the given sented in form of a rule, thus :- decimal by that number which Divide the lowest denomina- it takes of the next lower detion given, annexing to it one nomination to make one of this or more ciphers, as may be higher, and from the right necessary, by that number hand of the product point off which it takes of the same to as many figures for decimals make one of the next higher as there are figures in the denomination, and annex the given decimal, and so conquotient, as a decimal to that tinue to do through all the dehigher denomination; so con- nominations; the several num tinue to do, until the whole bers at the left hand of the shall be reduced to the deci- decimal points will be the mal required. value of the fraction in the proper denominations. EXAMPLES FOR PRACTICE.
EXAMPLES FOR PRACTICE. 3. Reduce 1 oz. 10 pwt. to the fraction of a pound.
'125 lb. Ans.
4. Reduce '125 lbs. Troy to integers of lower denomina tions.
12 oz. 1'500
pwt. 10'000. Ans. 1oz.10pwt.
5. Reduce 4 cwt. 2g qrs. to the decimal of a ton.
6. What is the value of 2325 of a ton?
Note. 23 2'6.
7. Reduce 38 gals. 3'52 qts, of beer, to the decimal of a hhd. of beer? hhd.
8. What is the value of "72
9. Reduce 1 qr. 2 n. to the decimal of a yard.
11. Reduce 17 h. 6 m. 43 sec. to the decimal of a day. 13.. Reduce 21 s. 10 d. to the decimal of a guinea.
15. Reduce 3 cwt. 0 qr. 7 lbs. 8 oz. to the decimal of a '15334821 of a ton?
10. What is the value of '375 of a yard?
12. What is the value of 713 of a day?
14. What is the value of 78125 of a guinea?
16. What is the value of
Let the pupil be required to reverse and prove the follow ing examples:
7. Reduce 4 rods to the decimal of an acre. 18. What is the value of "7 of a lb. of silver?
19. Reduce 18 hours, 15 m. 50'4 sec, to the decimal of a day.
20. What is the value of '67 of a league?
¶ 76. There is a method of reducing shillings, pence and farthings to the decimal of a pound, by inspection, more simple and concise than the foregoing. The reasoning in relation to it is as follows:
rb of 20 s. is 2 s.; therefore every 2 s. is, or '1 £. Every shilling is 80, or '05 £. Pence are readily reduced to farthings. Every farthing is £. Had it so happened, that 1000 farthings, instead of 960, had made a pound, then every farthing would have been Tobo, or '001 £. But 960 increased by part of itself is 1000; consequently, 24 farthings are exactly 1880, or '025 £., and 48 farthings are exactly 58, or '050 £. Wherefore, if the number of farthings, in the given pence and farthings, be more than 12, 24 part will be more than ; therefore add 1 to them: if they be more than 36, part will be more than 14; therefore add 2 to them: then call them so many thousandths, and the result will be correct within less than
of robe of a pound. Thus, 17 s. 5 d. is reduced to the
decimal of a pound as follows: 16 s. 8 £. and 1 s. = 05. Then, 5 d. 23 farthings, which, increased by 1, (the number being more than 12, but not exceeding 36,) is '024 £., and the whole is '874 £. the Ans.
Wherefore, to reduce shillings, pence and farthings to the decimal of a pound by inspection,—Call every two shillings one tenth of a pound; every odd shilling, five hundredths; and the number of farthings, in the given pence and farthings, so many thousandths, adding one, if the number be more than twelve and not exceeding thirty-six, and two, if the number be more than thirty-six.
77. Reasoning as above, the result, or the three first figures in any decimal of a pound, may readily be reduced back to shillings, pence and farthings, by inspection. Double the first figure, or tenths, for shillings, and, if the second figure, or hundredths, be five, or more than five, reckon another shilling; then, after the five is deducted, call the figures in the second and third place so many farthings, abating one when they are above twelve, and two when above thirty-six, and the result will be the answer, sufficiently exact for all practical purposes. Thus, to find the value of '876 £. by inspection :
'8 tenths of a pound
405 hundredths of a pound
'876 of a pound
EXAMPLES FOR PRACTICE.
1. Find, by inspection, the decimal expressions of 9 s. 7 d., and 12 s. 03 d. Ans. '479£., and '603£. 2. Find, by inspection, the value of '523 £., and '694 £. Ans. 10 s. 51 d., and 13 s. 10 d. 3. Reduce to decimals, by thspection, the following sums, and find their amount, viz.: 15 s. 3 d.; 8 s. 11 d.; 10 s. 61 d.; 1 s. 81 d.; 1 d., and 24 d. Amount, £1'833.
4. Find the value of '47 £.
Note. When the decimal has but two figures, after taking out the shillings, the remainder, to be reduced to thousandths. will require a cipher to be annexed to the right hand, or supposed to be so. Ans. 9 s. 4 d.