1. Position. 288. Position is a rule by which the true answer to a certain class of questions is discovered by the use of false or sup. poved numbers. 289... Supposing A's age to he dociloke that of B's, aud.B's age triple that of C's, and the sum of their ages to be 140 years; what is the age of each ? Let us suppose C's age to be 8 years, thou, by the questinn, B's age is 3 times 8=2+ years, and A's 2 times 24-48, and their sum is (8+24+48=1 80. Now, as the ratios are the same, both in the true and supposed ages, it is evident that the true sum of their ages will have the same ratio to the true age of each individual, that the sum of the supposed ages has to the supposed age of each individual, that is, 30 : 8 :: 140 : 12, C's true age; or, 80: 24 : : 1:10:42, B's age, or 80 : 48 : : 140 : 84, A's age. This operation is called Single Position, and may be expressed as follows: 290. When the result has the same ratio to the supposition that the given number has to the required one. Rull.-Suppose a number, and perform with it the operation described in the question. Then, by proportion, as the result of the operation is to the supposed number, so is the given result to the true number required. 2. What number is that, A vessel has 3 cocks ; which, being increased by 1, the first will fill it in 1 hour, and 1 itself, will be 125.? the second in 2, the third in Then 50 : 24 :: 125 : -60 Ans. 3; in what time will they all Or by fractions. fill it together? f=1 -12 Let 1 denote the Ans. 1 hour. 8 required number : 5. A person, after spending I= 6 | then and 4 of his money, had 1+1+1+t=125, $60 left; what had he at Result 50 or 13+++ first ? Ans. $144. =, and i 6. What number is that, H) 125 60 Ans. from which, if 5 be subtract(See p. 104, Miscel.) ed, f of the remainder will 3. What number is that be 40 ? Ans. 65. whose 6th part exceeds its 8th part by 202 Ans. 480. Sup. 24 II. When the ratio betroeen the required and the supposed num ber differs from that of the given number to the required one. 291. RULE.—Take any two numbers, and proceed with each according to the condition of the question, noting the 991. POSITION. 125 errors. Multiply the first supposed number by the last error, and the last supposed number by the first error; and if the errors be alike (that is, both too great or both too small), divide the difference of the products by the difference of the errors; but if unlike, divide the sum of the products 'by the sum of the errors, and the quotient will be the answer, NOTE.-This rule is founded on the supposition that the first error is no the second, as the difference between the true and first supposed is to the difference between the true and second supposed number; when that is ac the exact auswer to the questiou cannot be found by thus rule the case, 7. There is a fish, whose head is 10 inches long, his tail i as long as his head, and half the length of his body, and his body is as long as his head and tail both ; what is the length of the fish? Suppose the fish to be 40 inches long, then 40 Again sup. 60 X dody = 20 30 10-10 10 40 10 ....10 0 The above operation is called Dorible Position. The above question, and most others belonging to this rule, may be solved by fractions, thus : The body=5 of the whole length'; the tail=1 of 4+10= +10, and the head 10 : then ++10+10=the length; but +3, and 4–5=104-10–20 in. and 20X4=80 in. Ans, 2 What number is that | double that of the second ; but which being increased by its if it be put on the second, his $, its and 5 more, will be value will be triple that of the doubled? Ans, 20. first; what is the value of eache . 3 A gentleman has 2 hors- | horse ? of, and a saddle worth $50; Ans. Ist horse, $30, 20, $40 if the saddle be put on the 4. A and B lay out egnal Best borse, his value will be shares in trado: A gains $126, 126 PERMUTATION OF QUANTITIES. 292, 293, 294 and Bloses $87, then A's | than A, at the end of 4 years money is double that of B; | finds himself $100 in debt; what did each lay out? what is their income, and what Ans. $300. do they spend per annum ? 5. A and B have both the Ans. $125 their inc. per ann. samo income ; A saves one fifth of his yearly, but B, by spending $50 per annum more B spends $150 } per ann. Permutation of Quantities. 292. Permutation of Quantities is a rule, which enables us to deter. mine how many differeni ways the order 'or position of any giveu number of things may be varied. 293. 1. How many changes may be made of the letters in the word and ? The letter a can alone have only one position, a, denoted by 1, a and a can have two positions, an and nu, denoted by 1X2=2. The three letters, &, , and d, can, any two of them, leaving out the third, have two changes, 1X2, consequently when the third is taken in, there will be 1x2x3= changes, which may be thus expressed : and, udn, rida, nu, dun and dna, 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 numbers from 1 up to the given number, continually together, and the last product will be the answer required 2. How many days can 7 5. How many changes may persons be placed in a differ- | be rung on 12 bells, and how ent position at dinner? 5040. I long would they be in ringing, 3. How many changes may 1 supposing 10 changes to be be rung on 6 bells? rung in one minute, and the Ans. 720. year to consist of 365 days, 5 4. How many changes can hours and 49 minutes ? be made in the position of the Ans. 479001600 changes, 8 notes of music ? and 91 years, 26d. 22h. 41m. Ans. 40320. time. 295298 PERIODICAL DECIMALS. 127 Periodical Decimals. 295. The reduction of vulgar fractions to decimals (129) presents two cases, one in which the operation is terminated, as į=0.375, and the other in which it does not terminate, 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 di-visions, except in one of the series of the numbers, 1, 2, 3, &c. up to the divisor, when the number of divisions exceeds " that of this series, some one of the former remainders must recur, and consequently the partial dividends must 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 7+ (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 årst and last of the repeating figures, as 032432A. 297. The fractions which have 1 for a numerator, and any "number of O’s for the denominator, can have no significant figure in their periods except 1. Thus f=0.1111+: dy=0,010104. oto=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+=5, 0.3=}, &c. Again, as 0.010101, or 0.01, is the developement of stor 0.02= y, and so on, and in like manner of ty &c. Hence, 298. To reduce a periodical, or circulating decimal, to a vid gar fraction. Rule.Write down one period for a numerator, and as many nines for a denominator as the number of figures in a period of the decimal. 1. What is the vulgar frac- 5. Reduce 0.769230 to tbe tion of 0.18 ? form of a vulgar fraction. 6. What vulgar fraction is fraction equal to 0.138 ? Ans. 73= 9 x 13+8=125=numeratok 900=denominator. 3. Reduce 0.83 to the form -.0.138=138=, Ans. of a vulgar fraction. 7. What vulgar fraction is Here 0.8 is 8 tenths, and 3 equal to 0.53? is 3 Oths=# of 1 10th, or 1 8. What'is the least vulgar 30th; then to'=*to's / fraction equal to 0.5925 ? 4. Reduce 275463 to the 9 What finite number is form of a vulgar fraction. equal to 31.62 ? Ans. 3121. Ans. 33351 Ans. As =#, Ans. Ans. 24. 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 called canding ? When descending? What What kind of series do these frac. is meant by the extremes? The tions form? What is the ratio 1 means? When the first and last What is the rule for the multiplicaTerms are given, how do you find tion of duodeciinals? How are all the common difference? How the denominations less iban a foot to be number of terras 1 How the sum regarded ? of the series ? 3. What is Position ? What does * 2. What is a Geometrical Pro. it suppose when single ? Wber gression? What is an ascending double'? What kind of questions series ? a descending? may be solved by the fornier ? by What is the ratio ? When the tirst the latter ? term and the ratio are given, how 6. What is meant by the permu. do you find any other term? When tation of quantities? How do you the first and last term and the ratio find the number of perinutatious ? are given, how do you tind the sum Explain the reason. of the series ? 7. What is meant by a periodical 3. What is annuity? When is decinál? by a single repetend? it is arrears ? What does an annu- | By a eunpound repetend ? 'How is Ny at compound interest iurn? a repetend deuoted ? How is a poHow do you find the amount of an riodical decimal changed to immity at wompound interest ? equivalent sigar frávotion ? |