In reducing such fractions, if the division be continued, the same fig ures will recur again and again in the decimal fraction. These fractions are called Repeating or Circulating Decimals. The figures which repeat are called a Repetend. A Repetend is distinguished by two dots written over the first and last of the figures that repeat; thus, }}=.297297+=.297. 243 REDUCTION OF CIRCULATING DECIMALS TO COM MON FRACTIONS. It can be proved that the Repetend of a Circulating Decimal equals a fraction whose numerator is the repetend, and whose denominator is as many 9's as there are places in the repetend. Hence the RULE. To reduce a Circulating Decimal to a common fraction: Express the repetend as a common fraction having as many 9's for the denominator as there are figures in the repetend, and reduce. If any part of the decimal fraction does not repeat, annex the reduced repetend to it, and change the complex fraction thus obtained to a simple fraction. NOTE. — Circulating decimals may be added, subtracted, multiplied, and divided, by first reducing them to common fractions. Other processes might here be given, but the reasoning is too abstruse for an elementary treatise. Reduce .09 to a common fraction. ILL. EX., I. 244. To REDUCE COMPOUND NUMBERS TO DECIMAL FRACTIONS OF HIGHER DENOMINATIONS. ILL. EX., I. Reduce 2 d. 3 qr. to the decimal of a shilling. Since 4 qr. equal 1 d., there will be as many d. as qr., or 4 d., which equals .75 d.; this, with the 2 d. given, equals 2.75 d.; since 12 d. equals 1 shilling, there will be as many shillings as d., &c. ILL. EX., II. What is the value of 3 rds. 4 yds. 2 ft. in the decimal of a rod? From the above, we deduce the following RULE. To reduce compound numbers to decimal fractions of higher denominations: Divide the number of the lowest denomination by what it takes of that denomination to make one of the next higher; place the quotient as a decimal fraction at the right of that higher; so continue till all the terms are reduced to the denomination required. EXAMPLES. 1. Reduce 7 d. 3 qr. to the decimal of a £. Ans. £.03229+ 2. Reduce 3 da. 22 h. 4 m. 48 sec. to the decimal of a week. Ans. .56 wk 3. Reduce 5 cwt. 3 qr. 10 lb. to the decimal of a ton. 7. What is the value of 2 fur. 7 rd. 10 ft. expressed in the decimal of a mile? 8. What part of a ream is 15 quires 12 sheets? 9. What part of an acre is 3 R. 15 rd. 6 yd. 82 ft.? 10. Reduce 7 S. 8° 5′ 38′′ to the decimal of a great circle, eng For Dictation Exercises, see Key. 245. TO REDUCE DECIMAL FRACTIONS OF HIGHER DE NOMINATIONS TO WHOLE NUMBERS OF LOWER DENOMINA TIONS. ILL. EX. Reduce .13125 lbs. Troy to oz., &c. Since 12 oz. 1 lb., there will be 12 times as many ounces as pounds, 1.575 oz.; since 20 pwt. 1 oz., there will be 20 times as many pwt. as ounces, : 11.5 pwt.; since 24 gr. = 1 pwt., there will be 24 times as many grains as pwt., = 12 gr. Ans. 1 oz., 11 pwt., 12 gr. Hence the RULE. To reduce decimal fractions of higher denominations to whole numbers of lower denominations: Multiply the decimal fraction by what it takes of the next lower denomination to make a unit of the denomination of the given decimal, pointing off as in multiplication of decimals; so continue till the number is reduced as low as is required. 246. QUESTIONS FOR REVIEW. What are DECIMAL FRACTIONS? How are they generally writen? how read? How distinguished from whole numbers? indicates the denomination? Which figure What is the name of the first place at the right of the point? of the second? third? fourth? fifth? sixth ? Which is the place of thousandths? of millionths? of billionths? of trillionths? Give the rule for reading a decimal fraction. Read 7.05 as a mixed number; as an improper fraction. Read .20 and .21 so that they may be distinguished. Read .504 and 500.004. Is the value of a decimal fraction altered by annexing ciphers? What is changed? Why does the value remain the same? What is the effect of placing a cipher between the decimal fraction and the point ? Give the rule for writing decimal fractions. Rule for Addition; for Subtraction; for multiplying by 10, 100, 1000, &c.; for dividing by 10, 100, &c.; general rule for multiplication. off. Illustrate the rule by an example, and give the reason for pointing Give the rule for division of decimals. Perform an example to illustrate the rule, and explain. When the dividend does not contain the divisor what must be done? Rule for reducing common fractions to decimals. Illustrate and explain. Rule for reducing a decimal to a common fraction. explain. Illustrate and What fractions cannot be reduced wholly to the decimal form? What are they called? What are the repeating figures called? How is a repetend distinguished? Rule for reducing circulating decimals to common fractions. Rule for reducing a compound number to decimals of higher denom< inations. Illustrate. Rule for reducing decimals to whole numbers of lower der omina tions. Illustrate. 247. MISCELLANEOUS EXAMPLES. 1. What is the amount of 3.75 tons, .085 tons, 1.17§ tons, and 7 cwt. 3 qr.? 2. What will be the interest on $585, for 6 days, if the interest on $1 be $.001? 3. What is the amount of $75823 for 7 y. 3 m. 15 d., if the amount of $1 for the same time be $1.5104163? 4. At $1.33 a pair, how many cases of shoes, of 63 pairs each, can be bought for $936? 5. How many acres of land in a lot which is 105 rd. 4 yd. 14 ft. long, and 100.356 rd. wide? 6. Required the price of three boards at $.03 per sq. ft., the boards being of the following dimensions: 17.75 ft. by 1 ft. 3 in.; 15 ft. 10 in. by 1.37 ft., and 13.5 ft. by .9163 ft. 7. What is the amount due for the following? 5200 ft. of boards at $20 per M. 8. If 3 hhd. 42 gall. 22 qt. of molasses cost $92.64, what is the price per hhd.? 9. What is the cost of board for 7 y. 10 m. 18 d., at $200 per year? What cost 10. 9 gall. 3.4 qt. of vinegar, at $.12 per gall.? 16 12 doz. chairs, at $1.90 apiece? 17. A road 9 m. 3 fur. 12 rd. long, at $2475 18. 12520 oranges, at $23 per hundred? 19. 3 cwt. 40 lb. herring, at 12 s. 6d. per cwt.? 20. At $4 per bu. how many bu., pk. and qt. can be bought for $15.371? |