2 287. To multiply a number consisting of feet, inches, seconds, &c. 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. QUESTIONS FOR PRACTICE. 8. How many cords in a pile of 4 foot wood, 24ft. long, and 6ft. 4' high? Ans. 4 cords. 9. How many square yards in the wainscoting of a room 18ft. long, 16ft. & wide, and 9ft. 10 high? Ans. 75yd. 3ft. 6'. 10. How much wood in a cubic pile measuring 8ft. on every side? Ans. 4 cords. 11. How many square feet in a platform, which is 37 feet 11 inches long, and 23 feet 9 inches broad? Ans. 900ft. 6' 3". 12. How much wood in a load 8ft. 4in. long, 3ft. 9in. wide, and 4ft. 5in. high? Ans. 138ft. 0' 3". 13. How many feet of floor 6. How much wood in a loading in a room which is 28ft. 6ft. 7 long, 3ft. 5′ high, and 3ft. 8' wide? Ans. 82ft. 5' 8" 4". 7. What is the solid content of a wall 53ft. 6' long, 12ft. 3′ high, and 2ft. thick? Ans. 1310ft. 9'. 6in. long, and 23ft. 5in. broad? Ans. 667ft. 4' 6". 14. How many square feet are there in a board which is 15 feet 10 inches long, and 93 inches wide? Ans. 12ft. 10 4" 6"". 4. 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 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 times 8-24 years, and A's 2 times 24-48, and their sum is (8+2448=) 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, 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, which, being increased by, and itself, will be 125? Then 50: 24: 125: 60 Ans. Sup. 24 Or by fractions. Let 1 denote the required -12 then Result 50 4. A vessel has 3 cocks; the first will fill it in 1 hour, the second in 2, the third in 3; in what time will they all fill it together? Ans. hour. 5. A person, after spending and of his money, had 1+1+1+1=125, $60 left; what had he at first? or 12 +++ †)125 (60 Ans. (See p. 104, Miscel.) 3. What number is that whose 6th part exceeds its 8th part by 20? Ans. 480. Ans. $144. 6. What number is that, from which, if 5 be subtractof the remainder will ed, II. When the ratio between the required and the supposed number 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 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 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 not 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 Again sup. 60 40 X of +10=20 10 of +10=25 65 1st error 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 of the whole length; the tail=} of }+10={ +10, and the head 10: then 1+1+10+10-the length; but +, and 4--1--10+10-20 in. and 20×4-80 in. Ans. double that of the second; but if it be put on the second, his value will be triple that of the first; what is the value of each 2. What number is that | which being increased by its , its and 5 more, will be doubled? Ans. 20. 3. A gentleman has 2 hors-horse? es, and a saddle worth $50; if the saddle be put on the first horse, his value will be Ans. 1st horse, $30, 2d, $40 4. A and B lay out equal shares in trade: A gains $126, and B loses $87, then A's 5. A and B have both the same income; A saves one fifth of his yearly, but B, by spending $50 per annum more than A, at the end of 4 years per ann. per ann. 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 in the word and? The letter a can alone have only one position, a, denoted by 1, a and n can have two positions, an and na, denoted by 1X2-2. The three letters, a, n, 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=6 changes, which may be thus expressed: and, adn, nda, nad, dan 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 persons be placed in a different position at dinner? 5040. 3. How many changes may be rung on 6 bells? Ans. 720. 4. How many changes can be made in the position of the 8 notes of music? Ans. 40320. 5. How many changes may be rung on 12 bells, and how long would they be in ringing, supposing 10 changes to be rung in one minute, and the year to consist of 365 days, 5 hours and 49 minutes? Ans. 479001600 changes, and 91 years, 26d. 22h. 41m. time. Periodical Decimals. 295. The reduction of vulgar fractions to decimals (129) presents two cases, one in which the operation is terminated, as -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 divisions, 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+ (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 -0.1111+. -0.01010+ 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+=4, 0.3=f, &c. Again, as 0.010101, or 0.01, is the developement of 0.02, and so on, and in like manner of, &c. Hence, 298. To reduce a periodical, or circulating decimal, to a vulgar 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. |