292, 293, 294. PERMUTATION OF QUANTITIES. 181 and B loses $87, then A's | than A, at the end of 4 years money is double that of B; | finds himself $100 in debe ; what did each lay out? what is their income, and Ans. $300.. what do they spend per an num? 5. A and B have both the same income; A saves one Ang. $125 their inc. per ann. fifth of his yearly, but B, by A spends $100) spending $50 per annum more B spends $150 per app. Permutation of Quantities. 292. Permutation of Quantities is a rule, which enables us to deter mine how many different ways the order or position of any given number of things may be varied. 293. 1. How many changes may be made of the letters iu the word and ? The letter a, can alone have only one position, a, denoted by 1, a and B can have two positions, on and na, denoted by ] * 2 2. The three leiters, a, n and d can, any two of them, leaving out the third, have 2 chang el, 1 * 2, consequently when the third is taken in, ihere will be 1 X 213 6 changes, which may be thus expressed : and, adn nda, nad, d'an and daa, and the same may be shown of any number of things : Hence, 294. To find the number of permutations that can be made of a given number of different things. RULE.-Multiply all the terms of the natural series of purgbers from 1 up to the given number, continually together, and the last product will be the answer required. 2. How many days can 71 5. How many changes may persons be placed in a differ. | be rung on 12 bells, and home ent position at dinner ? 5040. Jong would they be in ridging, 3. How many changes may supposing 10 changes to be bo rung on 6 bells ? Ans. 720. | rung in one minute, and the 4 How many changes can ! year to consist of 365 daye, 5 be made in the position of the hours and 49 minutce ? Ans. 479001600 changes, ar 8 notes of music ? Ads. 40320. 1 9 years, 26d. 226.41m.'. Periodical Decimals. 295. The reduction of vulgar fractions to decimals (129) presents two cases, one in which the operation is terminated, as f=0.375, and the other in which it does not terininate, as =0.272727, &c. In fractions of this last kind, whose decimal value cannot be exactly found, it will be observed that the same figures return periodically in the same order. Hence they have been denominated periodical decimals. 296. Since in the reduction of a vulgar fraction to a decimal, there can be no remainder in the successive divisions, except in one of the series of the numbers, 1, 2, 3, &c. up to the divisor, when that number of divisions exceeds that of this series, some one of the former remainders must recur, and consequently the partial dividends inust return in the same order. The fraction =0.333+ Here the same figure is repeated continually, it is therefore called a single repetend. When two or more figures are repeated, as 0.2727+ (295) or 324324, it is called a compound repetend. A single repetend is denoted by a dot over the repeating figure, as 0.3 and a compound repetend by a dot over the first and last of the repeating figures, as 0324324. 297. The fractions which have 1 for a numerator, and any number of 9's for the denominator, can have no significant figure in their periods except 1. Thus 30.1111t. ab=0.01010 . o&r=0.001001001 This fact enables us easily to ascertain the vulgar fraction from which a periodical decimal is derived. As the 0.1111+ is the developement of }, 0.22+= 0.3=$, &c. Again, as 0.010101, of 0.01 is the developement of glas, 0.02=5, and so on, and in like manner of gtg, &c. Hence 298. To reduce a periodical, or circulating decimal, to a vulgar fraction. - 298. Rule-Write down one period for a numerator, and as maay nines for a denominator as the number of figures in a period of the decimal. 1. What is the vulgar frac. | is 3 9ths=1 of 1 10th, or 1 tion of 0.18? | 30th; then to ti'o =.+ Ans. jf=1t: 1 =, Ans. 2. Reduce 0.72 to a vulgar 4. Reduce 275463 to the fraction. | form of a vulgar fraction. Ans. zzril Ans. 3953 3. Reduce 0.83 to the form 5. Reduce 0.769230 to the of a vulgar fraction. form of a vulgar fraction.' Here 0.8 is 8 tenths and 3 Ans. 19. REVIEW 1. What is an Arithmetical Pro- ' 4. What is the common division gression ? When is the series as- of a foot ? What are these calscending? When descending?! ed? What kind of series do these What is meant by the extremes ? fractions form? What is the ratio ? The means? When the first and What is the rule for the multipli bast terms are given, how do you i cation of duodecimals? How are find the common difference? How all denominations less than a foot the number of terms? How the to be regarded ? sum of the series? 5. What is Position? What does 2. What is a Geometrical Pro- ; it suppose when single? When gression ? What is an ascending double? What kind of questions series? What a descending? I may be solved by the former ? by What is the ratio ? When the first the latter? term and the ratio are given. how 6. What is ineant by the permu. do you find any other term? When Lalion of quantities? How do you the first and last term and the ratio find the number of permulations? are given, how do you find the sumn Explain the reason. of the series? 7. What is meant by a periodical 3. What is annuity? When is | decimal ? By single repetend ? it in arrears? What does an annu- | By a compound repetend? How is ity al compound interest form ? ; a repelend denated ? How is a peHow do you And the annount of an rivaical deciinal changed to an annuity al compound interest ? I equivalent vulgar fraction ? 302. EXCHANGE OF CURRENCIES. 185 RULE.Reduce the shillings, &c. to the decimal of a pound; then if it is English currency, divide by 0.225, if Capada, by 0.25, if N. E. by 0.3, if N. Y. by 0.4, if Penn. by 0.375, and if Georgia, by 0:23;--the quotient will be their value in dollars, cents and mills. And to change Federal Mopey into the above currencies, multiply it by the preceding decimals, apd the product will be the answer in pounds and decimal parts. |