tion,-Annex one or more ciphers, as may be necessary, to the namerator, and divide it by the denominator If then there ha a remainder, annex another cipher, and drival us v.jure, and vo 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, 4, 4, and 2 to decimals. Ans. '5; 25; '025; '00797 +. 5. Reduce, Tobo, 1755, and be to decimals. Ans. 692; '003; '0028+; '000183 +. 6. Reduce 179,367, 680 to decimals. 7. Reduce,,,,,,, go to decimals. 8. Reduce, , §, 1, †, t, t, zb, 25, 7 to decimals. If be reduced to a decimal, it produces '001; consequently, 0025, and '037 = 3, and 425 33, &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? 014? 324? Ans. 78. '01021 ? In this fraction, the repetend begins in the second place, or place of hundredths. The first figure, 4, is fo, and the repetend, 3, is of To, that is, ; these two parts must be added together. 3 38 3, Ans. Hence, to find the value of a mixed repetend,—Find the value of the two parts, separately, and add them together. 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, (T 63,) like integers, are To reduce a compound num-| To reduce the decimal of a 1. Reduce 7 s. 6 d. to the decimal of a pound. 2. Reduce 375 £, to integers of lower denominations. 375 £. reduced to shillings, 6 d. reduced to the decimal 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 numtinue 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. EXAMPLES FOR PRACTICE. 3. Reduce 1 oz. 10 pwt. to the fraction of a pound. OPERATION. 20)10'0 pwt. 12)1'5 oz. '125 lb. Ans. N value of the fraction in the EXAMPLES FOR PRACTICE. 4. Reduce '125 lbs. Troy to integers of lower denomina tions. OPERATION. lb. '125 12 oz. 1'500 20 pwt. 10'000. Ans. 1oz. 10pwt. . 5. Reduce 4 cwt. 2g qrs. to the decimal of a ton. Note. 23 = 2'6. 7. Reduce 38 gals. 3'52 qts. of beer, to the decimal of a hhd. 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. 17 lbs. 8 oz. to the decimal of a ton. 6. What is the value of 2325 of a ton? 8. What is the value of "72 hhd. of beer? 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 '15334821 of a ton? Let the pupil be required to reverse and prove the following examples: 17. Reduce 4 rods to the decimal of an acre. 18. What is the value of " 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? 21. Reduce 10 s. 94 d. to the fraction of a pound. ¶ 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: 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, or '001 £. But 960 increased by part of itself is 1000; consequently, 24 farthings are exactly 1880, or '025 £., and 48 farthings are exactly 88, or '050 £. Wherefore, if the number of farthings, in the given pence and farthings, be more than 12,4 part will be more than ; therefore add 1 to them: if they be more than 36, 4 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 Toby of a pound. Thus, 17 s. 5 d. is reduced to the 6 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 : an of a pound ndths, abating 1, 25 farthings= pound 0 s. 61 d. EXAMPLES FOR PRACTICE. Find, by inspection, the decimal expressions of 9 s. 7 d., d 12 s. Og d. Ans. 479 £., and '603£. . Find, by inspection, the value of '523 £., and '694 £. Ans. 10 s. 51 d., and 13 s. 104 d. $3. Reduce to decimals, by inspection, the following sums, ad find their amount, viz.: 15 s. 3 d.; 8 s. 11 d.; 10 s. Amount, £1'833. d. ; 1 s. 84 d. ; † d., 4. Find the value of '47 £. and 21 d. Note. When the decimal has but two figures, after taking it the shillings, the remainder, to be reduced to thousandths, ill require a cipher to be annexed to the right hand, or Ans. 9 s. 4 d. supposed to be so. 5. Value the following decimals, by inspection, and find their amount, viz.: 785 £.; '357 £.; '916 £.; "74 £.; '5 £.; '25 £.; '09 £.; and '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 decimal 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 the 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 hard, how do they affect their value? 10. How are decimals read? 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 fracti on? 15. What is observed respecting the denominations in federal money? 16. What is the rule for addition and subti action of decimals, particularly as respects placing the dec imal point in the results? -multiplication? divisioon? 17. How is a common or vulgar fraction reduced to a de mal? 18. What is the rule for reducing a compound nun ber to a decimal of the highest denomination contained it? 19. What is the rule for finding the value of any givel decimal 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? |