« ΠροηγούμενηΣυνέχεια »
ly 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:-Po reduce a common to a decimal fraction,-Annex one or more ciphers, as may be necessary, to the numerator, and divide it by the denominator. If then there be a remainder, annex another cipher, and divide as before, 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 , d, 12., and
Ans. '5; 625; 025 ; 00797 +. 6. Reduce }, TOO785
Ans. -692 +; '003; 0028 +; '000183 to 6. Reduce , 367, st. to decimals. 7. Reduce , oy, gg, }, }, , iri gig to decimals. 8. Reduce 8, 8, 8, 5, , , , , , s to decimals.
petend. If other figures arise before those which circulate, as '743333, '143. 010101, &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 39.
A compound repetend 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 f to a decimal, we obtain a continual repetition of the figure 1; thus, '11111, that is, the repetend'l. The value of the repetend '1, then, ist; the value of “222, &c., the repetend “ż, will evidently be twice as much, that is, f. In the same manner,
š, and so on to 9, which == 1. 1. What is the value of 8?
g 2. What is the value of ? Ans. g = What is the value ofis ? of ? of «i ? - of 5 ? of .g'?
of i? If be reduced to a decimal, it produces ‘010101,, or the repetend öi. The repetend 'oż, being 2 times as much, must be on, and =g, and ise being 48 times a much, must be 18, and 674
REDUCTION OF DECIMAL FRACTIONS.
175. Fractions, we have seen, (1 63,) like integers, are re duced from low to higher denominations by division, and from high to lower denominations by multiplication.
To reduce a compound number To reduce the decimal of a to a decimal of the highest deno-Thigher denomination to integers mination.
of lower denominations. 1. Reduce 7 s. 6 d. to the deci. 2. Reduce "375 £. to integers mal of a pound.
of lower denominations. 6 d. reduced to the decimal of '375 £. reduced to shillings, a shilling, that is, divided by 12, that is, multiplied by 20, is 750 is '5 s., which annexed to the 1 s.; then the fractional part, '50 s. making 75 s., and divided by s., reduced to pence, that is, mul20, is '375 £. the Ans.
tiplied by 12, ís 6 d. The process may be presented
Ans. 7 s. 6 d. in form of a rule, thus :-Divide That is,--Multiply the given the lowest denomination given, decimal by that number which it annexing to it one or more ci- takes of the next lower denominaphers, as may be necessary, by tion to make one of this higher, that number which it takes of the and from the right hand of the same to make one of the next product point off as many figures higher denomination, and annex lor decimals as there are figures the quotient, as a decimal, to that in the given decimal, and so contihigher denomination; so conti- nue to do through all the denominue to do, until the whole shall nations; the several numbers at
and 037 =
and 425 =
If ogo be reduced to a decimal, it produces coji; consequently «002
As this principle will apply to any number of places, we have this general Rule for reducing a circu. lating decimal to a vulgar fraction,-Make the given repetend the nu. merator, and the denominator will be as many 9s as there are repeating figures. 3. What is the vulgar fraction equivalent to 0704?
999 4. What is the value of 1003 ? 014? 32,3 - 101021 ? 12463 ? 4002103 ?
Ans. to last, 701 5. What is the value of 43 ?
In this fraction, the repetend begins in the second place, or place of hun dredths. The first figure, 4, is to, and the repetend, 3, is g of it 90 Ans. Hence, to find the value of a mixed repetend,-Find the value of the two parts, separately, and add them together. 6. What is the value of '153 ? Toy + gio=138= 23
Ans. 7. What is the value of 0041 ?
. 8. What is the value of 138? (16? 4iz?
It is plain, that circulates may be added, subtracted, multiplied, and divi. ded, by first reducing them to their equivalent vulgar fractions.
po; these two parts must be added together. 1 + 0 = * = 18
be reduced to the decimal requir- the left hand of the decimal points ed.
will be the value of the fraction
in the proper denominations. EXAMPLES FOR PRACTICE. EXAMPLES FOR PRACTICE.
3. Reduce 1 oz. 10 pwt. to the 4. Reduce ‘125 lbs. Troy to in. fraction of a pound.
tegers of lower denominations. OPERATION
OPERATION. 20 ) 100 pwt.
12 12 ) 145
oz. 1'500 (125 lb. Ans.
pwt. 10ʻ000 Ans. 1 oz. 10 pwt. 6. Reduce 4.cwt. 23 yrs. to 6. What is the value of '2325 the decimal of a ton.
of a ton ? Note. 23 = 26.
7. Reduce 38 gals. 3*82 qts. of 8. What is the value of 672 hhd. beer to the decimal of a hhd. of beer ?
9. Reduce 1 qr. 2 n. to the de 10. What is the value of "375 cimal of a yard.
of a yard? 11. Reduce 17 h. 6 m. 43 sec. 12. What is the value of 1713 to the decimal of a day.
of a day? 13. Reduce 21 s. 103 d. to the 14. What is the value of 578125 decimal of a guinea.
of a guinea ? 15. Reduce 3 cwt. O qr. 7 lbs. 16. What is the value of '158 oz. to the decimal of a ton. 334821 of a ton ?
1 Let the pupil be required to reverse and prove the following esamples :
17. Reduce 4 rods to the decimal of an acre.
176. 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 fol.
shilling is zo
to of 20 s. is 2 s.; therefore, every 2 s. is , or '1 £. Every
Tégs or ‘05 £. Pence are readily reduced to far. things. Every farthing is obo £. Had it so happened, that 1000 farthings, instead of 960, had made a pound, then every farthing would have been Tobo, or, 6001 £. But 960 increased by a part of itself is 1000 ; consequently, 24 farthings are exactly --25 s, or, '025 £., and 43 farthings are exactly or '050 £. Wherefore, if ine number of farthings, in the given pence and farthings, be more
than 12, z part will be more than $; therefore, add 1 to them, if they be more than 36, 24 part will be more than 1$; therefore, add 2 to them: then call them so many thousandths, and the result will be correct within less than } of bo-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, 54 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 farthing's, in the given pence and farthings, so many thousandths, adding one if the number be more than twelve and not exceeding thirtysix, and two if the number be more than thirty-six.
177. 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 thirtysix, and the result will be the answer, sufficiently exact for all practical purposes. Thus, to find the value of '876 £. by inspection : 68 tenths of a pound
= 16 shillings. '05 hundredths of a pound
1 shilling: *U26 thousandths, abating 1, = 25 farthings
64 d. (876 of a pound
= 17 s.
Ans. EXAMPLES FOR PRACTICE. 1. Find, by inspection, the decimal expressions of 9 s. 7 d., and 12 s. 08 d.
Ans. '479 £., and '603 £. 2. Find, by inspection, the value of '523 £., and '694 £.
Ans. 10 s. 55 d, and 13 s. 10} d. 3. Reduce to decimals, by inspection, the following sums, and find their amount, viz. 15 s. 3 d. ;8.6: 114 d.; 10 s. 64 d.; 1 s. 83 d. ; $ d., and 24 d.
Amount, £i'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. 48 d. 5. Value the following decimals, by inspection, and find their amount, viz. "785 £. ; '357 £. ; '916 £. ; '74 £.; '5 £.; "25 £. ; '09 £.; aid '008 £.
Ans. 3 £. 12 s. 11 d.
SUPPLEMENT TO DECIMAL FRACTIONS.
QUESTIONS. 1. What are decimal fractions ? 2. Whence is the term derived ? 3. How do decimal differ from common fractions? 4. How are de imal fractions written ? 5. How can the proper denominator to a decimal fraction be known, if it be not expressed? 6. How is the value of every figure determined ? 7. What does the first figure on ihe right hand of the decimal point signify? the second figure?
third figure? fourth figure? 8. How do ciphers, placed at the right hand of decimals, affect their value? 9. Placed at the left hand, how do they afiect their value? 10. How are decimals read? 11. How are decimal fractions, having different denominators, reduced to a common denominator? 12. What is a mixed number? 13. How may any whole number be reduced to decimal parts? 14. How can any mixed number be read together, and the whole expressed in the form of a common fraction ? 15. What is observed respecting the denoininations in federal money? 16. What is the rule for addition and subtraction of decimals, particularly as respects placing the decimal points in the results ?
multiplication ? division ? 17. How is a common or vulgar fraction reduced to a decimal ? 18. What is the rule for reducing a compound number to a decimal of the highest denomination contained in it? 19. What is the rule for finding the value of any given decinal of a higher denomination in terms of a lower ? 20. What is the rule for reducing shillings, pence and farthings to the decimal of a pound, by inspection ? 21. What is the reasoning in relation to this rule? 22. How may the three first figures of any decimal of a pound be reduced to shillings, pence and farthings, by inspection ?
EXERCISES. 1. A merchant had several remnants of cloth, measuring as fol. lows, viz. : 7% yds. How many yards in the whole, and what would the
whole come to at $3'67 per yard ? CS 16
Note. Reduce the common fractions to decimals. 93
Do the same wherever they occur in the examples which
Ans. 36'475 yards. $133'863 +, cost. 3.5......
2. From a piece of cloth, containing 365 yards, a merchant sold, at one time, 7-35 yds., and, at another time, 12; yds. ; how much of the cloth had he left?
Ans. 1667 yds. 3. A farmer bought 7 yards of broadcloth for 8 for £.; a barrel of flour for 24 £.; a cask of lime for 18 £., and 7 lbs. rice for £.; he paid i ton of hay, at 376 £., i cow at of £., and the l'alance in pork att
L£. per lb.; how many were the pounds of pork ?