282. A person bought 6 brooms, giving 3 cents for the first and 96 cents for the last, and the prices form a geometrical series, the ratio of which was 3; what was the cost of all the brooms? The price would be the sum of the following series : 3+6+12+24 for 48+96=189 cents, Ans. If the foregoing series he multiplied by the ratio. 2. the product is 6+12+24+48 +96=192 whose siim is twice that of the first. Now subtracting the first series from this, the remainder is 192 -3=139=ihe sum of the first series. Had the ratio been other than 2. the remainder would have been as many times the sum of the series as the ratio, less 1, and the renrainder is always the difference between the first term and the product of the last term by the ratio. Hence, II. The first and last term and ratio given to find the sum of the series. RULE.--Multiply the last term by the ratio, and from the product subtract the first term, the remainder divided by the ratio, less 1, will give the sum of the series.. 2. The first term of a geo. / second, $4 the third, and so on, metrical series is 4, the last each succeeding payment beterm 972, and the ratio 3; ing double the last; and wbat what is the sum of the series ? will be the last payment? $4095 tbe debt. 3 -- 13972M3 --4=1456 Ans. Ans: $2048 last pay't. NOTE.-The marks drawn uver | the numbers show that 4 must be 5. & gentleman being asktaken fron the product of 972 by 3, ed to dispose of a horse, said and the remainder divided by (3–1 he would sell him on condition ==> 2. This mark is called a vincu. I of having 1 cent for the first bum. . nail in his shoes, 2 cents for 3. The extremes of a geo. the second, 4 cents for the inetrical progression are 1024 third, and so on, doubling the and 59049, and the ratio 1; what is the sum of the series ? ! price of every nail to 32, the : number of pails in his four Ans. 175099. shoes ; what was the price of 4. What debt will be dis. the horse at that rate ? charged in 12 months by pay. | Ans. $42949672.95.. ing $1 the first month, $2 the 283. Ifa pension of $100 dollars per annuin be forboni 6 years, what is there due at the end of that time, allowing compound interest at 6 per cent. ? Whatever the time, it is obvious that the last year's: peosion will draw no interest, it is therefore only $100; the last but one will draw interest one year, amounting to $106.; the last but two, interest (compound) for 2 years, amounting to $112,36; and so on, forming a geometrical' progres. sion, whose first term is 100, the ratio 1.06, and the sum of this series will be the amount due. To find the last term (281) say, 1.065* 100% 133.82255776, the sixth term ; and to find the sum of the series (282) say, 133.82255777 1.06-100=41.8519112256, wbich divided by 1.06. =. @G, gives $667.5318576 Ang, or sum Jue. 284, 285, 286. DUODECIMALS. 177 - 284. A sum of money payable every year for a number of years is called annuity. When the payment of an annuity is forborn, it is said to be in arrears. 1. What is the amount of an an. 1 2. Ifa yearly rent of $50 he fornuity of $10, to continue 5 years, | born 7 years, to what does it amount allowing 5 per cent. compound inte. at 4 per cent. compound interest? rest? Ans. $221.025. Ans. 9394.91. 3. Duodecimals. 285. Of the various subdivisions of a foot, the following is one of the most common. TABLE. 1 foot is 12 inches, or primes, ' 1= ift. 1 inch 12 seconds " = 'te 1 second 12 thirds " of ta= IT. 1 third 12 fourths ""t's of It's of 12 =178&.. forming a decreasing geometrical progression, whose first terna is 1, and ratio 12. Hence they are called Duodecimals. NOTR.-When feet are concerned, the product is of the same cienomi. nation as the term multiplying the feet ; and when feet are not concerra ord, the name of the product will be denoted by the sum of the indices of the two factors, or strokes over thein. Thus 4'*7'8". 'Therefore, 287. To multiply a number consisting of feet, inches, seconds, fc. by another of the same kind. Rule.--Write the several terms of the multiplier under the corresponding terms of the multiplicand; then multiply the whole multiplicand by the several terms of the multiplier successively, beginning at the right hand, and placing the first term of each of the partial products under its respective multiplier, remembering to carry one for every 12 from a lower to the next higher denomination, and the sum of these partial products will be the answer, the left hand term being feet, and those towards the right primes, seconds, &c. This is a very useful rule in measuring wood, boards, &c. and for artificers in finding the contents of their work. 1. Position. 288. Position is a rule by which the true answer to a cer. taia class of questions is discovered by the use of false, or supposed numbers. 289. Supposing A's age to be double that of B's, and 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, then by the question B's age is 3 . times 8=24 years, and A's 2 times 24548, and their sum is (8-424-4-48 F) 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 saine 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 80:8;:140:12 C's true age; or 80 : 24 : : 140 : 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. RULE.-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, 8th part by 20? Aps. 480. which, being increased by 4. A vessel has 3 cocks, 1, $ and itself, will be the first will fill it in 1 hour, 125 ? the second in 2, the third Then 50: 24 ::125 :60 Ans. l in 3; in what time will Sup. 24 Or by fractions. I they all fill it together? {–12 Let 1 denote the Ans. 1 hoạr. $= 8 required number: 1. : 1 . 5. A person, after spend*= 6 then - ++4+3+4=125, ing $ and of his money, - result 50 or 14+ had' 860 left'; what had he | at first? M= Ans. $144. and is . ) 125 (60 Ans. ' 1 6. What number is that, (See p. 159, Miscel.) ; from which, if 5 be: sub. 3. What number is that tracted, 'of the remainder whose 6th part exceeds its / Will lite will be 40 ? Ans. 65., II. When the ratio between the required and the supposed num. der differs from that of the given number to the required one. 21. RULE. --Take any two numbers, and proceed with each socorling to the condition of the question, noting the errors. Multiply the first supposed number by the last er. ror, 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 differeace of the products by the difference of the errors; but if unlike, divide the sum of the products by the sum of tbe errors, and the quotient will be the answer. NOTE.-This rule is founded on the supposition that the first error is to 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 got the case, the exact answer to the question cannot be found by this rule. 7. There is a fish whose head is 10 inches long, his tail is 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 40 10 The above operation is called Double Position. The above question, and most others belonging to this rule, may be solved by fractions, thus : the body, 4 of 1=4+10=the tail, and 10=the head, and 1 +* +10+10=the length 4+1=4 and 4-*== 10410=20 and 20 X4=30 Ans. 2. What number is that I double that of the second ; but which being increased by | if it be put on the second, bis its 1. its tand 5 more. will | value will be triple that of the be doubled ? Ans. 20. | first; what is the value of each horse? 3. A gentleman has 2 hor- ! Aps. 1st horse $30, 2d $40. ses, and a saddle worth $50; } if the saddle be put on the 4. A and B lay out equal first horse, his value will be shares in trade: Agains $126, |