CASE IV. 347. Decimal Compound Numbers reduced to whole ones. 1. Reduce £.387 to shillings, pence and farthings. Operation. 20 shil. 7.740 12 pence 8.880 4 far. 3.520 Ans. 7s. 8d. 3 far. answer. Hence, Multiply the given decimal by 20, because 20s. make £1, and point off as many figures for decimals, as there are decimal places in the multiplier and multiplicand. (Art. 324.) The product is in shillings and a decimal of a shilling. Then multiply the decimal of a shilling by 12, and point off as before, &c. The numbers on the left of the decimal points, viz: 7s. 8d. 3 far., form the 348. To reduce a decimal compound number to whole numbers of lower denominations. Multiply the given decimal by that number which it takes of the next lower denomination to make ONE of this higher, as in reduction, and point off the product, as in multiplication of decimal fractions. (Art. 324.) Proceed in this manner with the decimal figures of each succeeding product, and the numbers on the left of the decimal point of the several products, will be the whole number required. 2. Reduce £.725 to shillings, pence and farthings. 3. Reduce £.1325 to shillings, &c. 4. Reduce .125s. to pence and farthings. 5. Reduce .825s. to pence and farthings. 6. Reduce .125 cwt. to pounds, &c. 7. Reduce .435 lbs. to ounces and drams. 8. Reduce .275 miles to rods, &c. 9. Reduce .4245 rods to feet, &c, 10. Reduce .1824 hhds. to gallons, &c. 13. Reduce .845 hr. to minutes and seconds. QUEST. 348. How are decimal compound numbers reduced to whole ones of a lower denomination? SECTION X. PERIODICAL, OR CIRCULATING DECIMALS.* ART. 349. Decimals which consist of the same figures or set of figures repeated, are called PERIODICAL, OR CIRCULATING DECIMALS. (Art. 339.) 350. The repeating figures are called periods, or repetends. If one figure only repeats, it is called a single period, or repetend; as .11111, &c.; .33333, &c. When the same set of figures recurs at equal intervals, it is called a compound period, or repetend; as .01010101, &c.; .123123123, &c. 351. If other figures arise before the period commences, the decimal is said to be a mixed periodical; all others are called pure, or simple periodicals. Thus .42631621, &c., is a mixed periodical; and .33333, &c., is a pure periodical decimal. OBS. 1. When a pure circulating decimal contains as many figures as there are units in the denominator less one, it is sometimes called a perfect period, or repetend. (Art. 344.) Thus, .142857, &c., and is a perfect period. 2. The decimal figures which precede the period, are often called the terminate part of the fraction. 352. Circulating decimals are expressed by writing the period once with a dot over its first and last figure when compound; and when single by writing the repeating figure only once with a dot over it. Thus .46135135, &c., is written .46135 and .33, &c., .3. 353. Similar periods are such as begin at the same place before or after the decimal point; as .i and .3, or 2.34 and 3.76, &c. Dissimilar periods are such as begin at different places; as .123 and .42325. Similar and conterminous periods are such as begin and end in the same places; as .2321 and 1634. * Should Periodical Decimals be deemed too intricate for younger classes, they can be omitted till review. REDUCTION OF CIRCULATING DECIMALS. CASE I.—To reduce pure circulating decimals to common fractions. 354. To investigate this problem, let us recur to the origin of circulating decimals, or the manner of obtaining them. For example, .11111, &c., or .i; therefore the true value of .11111, &c., or .i, must be from which it arose. For the same reason .22222, &c., or .2, must be twice as much or ; (Art. 186;) .33333, &c., or .3=3; .4=4; .5=5, &c. 9 999999 Again, .010101, &c., or .01; consequently .010101, &c., or .01; .020202, &c., or .02=&; .030303, &c., or .03=3; .070707, &c., or .07=, &c. So also.001001001, &c., or .001; therefore .001001, &c., or .001; .002; &c. In like manner =.142857; (Art. 337;) and 142857142857; for, multiplying the numerator and denominator of by 142857, we have 5. (Art. 191.) So is twice as much as;, three times as much, &c. Thus it will be seen that the value of a pure periodical decimal is expressed by the common fraction whose numerator is the given period, and whose denominator is as many 9s as there are figures in the period. Hence, 355. To reduce a pure circulating decimal to a common fraction. Make the given period the numerator, and the denominator will be as many 9s as there are figures in the period. Ex. 1. Reduce .3 to a common fraction. Ans. 3, or 1. Ans. §, or . CASE II. To reduce mixed circulating decimals to common fractions. 356. 11. Reduce .16 to a common fraction. Analysis. Separating the mixed decimal into its terminate and periodical part, we have .16 .1+.06. (Art. 320.) Now .1=; (Art. 312;) and .06; for, the pure period .6, (Art. 351,) and since the mixed period .06, begins in hundredths' place, its value is evidently only as much; but ÷10=. (Art. 227.) Therefore .16=+. Now and, reduced to a common denominator and added together, make 15, or 1. Ans. OBS. In mixed circulating decimals, if the period begins in hundredths' place it is evident from the preceding analysis that the value of the periodical part is only as much as it would be, if the period were pure or begun in tenths' place; when the period begins in thousandths' place, its value is only as much, &c. Thus .6; .06 +10=; .006 §÷100=,, &c. part 357. Hence, the denominator of the periodical part of a mixed circulating decimal, is always as many 9s as there are figures in the period with as many ciphers annexed as there are decimals in the terminate part. 12. Reduce .8567923 to a common fraction. 100 Solution.-Reasoning as before .8567923-15+597933. Reducing these two fractions to the least common denominator, (Art. 261.) 99999-8499915 whose denominator is the same as that of the other. Now 8493915+5933300 8567838 Ans. Contraction. 8500000 85 8499915 1st Nu. 67923 2d Nu. 8567838 9999900 Ans. - = 9999900. To multiply by 99999, annex as many ciphers to the multiplicand as there are 9s in the multiplier, &c. (Art. 105.) This gives the numerator of the first fraction or terminate part, to which add the numerator of the second or periodical part, and the sum will be the numerator of the answer. The denominator is the same as that of the second or periodical part. Second Method. 8567923 the given circulating decimal. 85 the terminate part which is subtracted. 8567838 the numerator of the answer. 8567838 Ans. 9999900 Note.-1. The reason of this operation may be shown thus: 8567923= 8500000+67923. Now 8500000-85-67923 is equal to 8567923-85. 2. It is evident that the required denominator is the same as that of the periodical part; (Art. 357;) for, the denominator of the periodical part is the least common multiple of the two denominators. Hence, 358. To reduce a mixed circulating decimal to a common fraction. Change both the terminate and periodical part to common fractions separately, and their sum will be the answer required. Or, from the given mixed periodical, subtract the terminate part, and the remainder will be the numerator required. The denominator is always as many 9s as there are figures in the period with as many ciphers annexed as there are decimals in the terminate part. Ans. 125, or 5. PROOF.-Change the common fraction back to a decimal, and if the result is the same as the given circulating decimal, the work is right. 13. Reduce .138 to a common fraction. 14. Reduce .53 to a common fraction. 15. Reduce .5925 to a common fraction. 16. Reduce .583 to a common fraction. 17. Reduce .0227 to a common fraction. 18. Reduce .4745 to a common fraction. 19. Reduce .5925 to a common fraction. CASE III.-Dissimilar periodicals reduced to similar and conter minous ones. 359. In changing dissimilar periods, or repetends, to simila and conterminous ones, the following particulars require attention. 1. Any terminate decimal may be considered as interminate by annexing ciphers continually to the numerator. Thus .46= .460000, &c.=.460. |