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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 thosc 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. 2. How much wood in a load 8. How many cords in a pile 7ft. 6 long, 4ft. 8' wide, and of 4 foot wood, 24ft. long, and 4ft. high?
6ft. 4' high? Ans. 140ft. or 1 cord 12ft.
Ans. 4 cords. Multiply the length by the width, 9. How many square yards and this product by the height. 3. How many square feet in 18ft. long, 16ft
. & wide, and
in the wainscoting of a room a board 16ft. 4in. long, and 2ft. 9ft. 10 high? 8in. wide ? Ans. 43ft. 6in. 811.
Ans. 75yd. 3ft. 6.
10. How much wood in a 4. How many feet in a stock cubic pile measuring 8ft. on of 12 boards 14ft.
side ? Ans. 4 cords. 1ft. 3 wide ? Ans. 217ft. 6'.
11. How many square feet Note.-Inches, it will be recol- in a platform, which is 37 feet lected, are so many. 12ths of a foot, 11 inches long, and 23 feet 9 whether the foot is lineal, square, inches broad? or solid. 6in, in the above answer
Ans. 900ft. 6 3". is of a square foot, or 72 square inches.
12. How much wood in a 5. What is the content of a load 8ft. 4in. long, 3ft. 9in. wide, ceiling 43ft. 3 long, and 25ft. and 4ft. 5in. high? 6 broad?
Ans. 138ft. O 3. Ans. 1102ft. 10 6'.
13. How many feet of floor6. How much wood in a load ing in a room which is 28ft. 6ft. 7' long, 3ft. 5' high, and in. long, and 23ft. 5in. broad? 3ft. 8' wide
Ans. 667ft. 4! 6". Ans, 82ft. 5' 8" 4". 14. How many square feet 7. What is the solid content are there in a board which is of a wall 53ft. 6 long, 12ft. 3 15 feet 10 inches long, and 93 high, and 2ft. thick ?
inches wide ? Ans. 1310ft. 9.
Ans. 12ft. 10 4" 6".
0. 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 each ?
Let us suppose C's age to be 8 years, then, by the estion, B's age is 3 times 8=24 years, and A's 2 times 2448, 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, 4. A vessel has 3 cocks ; which, being increased by }, the first will fill it in 1 hour, 4 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?
Ans. A hour.
fand of his money, had 1+1+1+1=125, $60 left; what had he at Result 50 or iftitut first ?
Ans. $144. =45, and 1=
6. What number is that, 3) 125 60 Ans.
from which, if 5 be subtract(See p. 104, Miscel.) ed, of the remainder will 3. What number is that be 40 ?
Ans. 65. whose 6th part exceeds its 8th part by 20? Ans. 480.
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
Sup. 24 +=12
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
40 Again sup. 60 40 10 body da
30 tail" of 4+10=20 1 of +10=25 60 5 head 105
10—10 10 40
0 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 1+10=1 +10, and the head 10 : then +*+10+10=the length; but +=l, and 4--==10+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 1, its ļ and 5 more, will be value will be triple that of the doubled ?
Ans. 20. first; what is the value of each 3. A gentleman has 2 hors- horse ? es, and a saddle worth $50; Ans. 1st horse, $30, 2d, $40 if the saddle be put on the 4. A and B lay out equal first horse, his value will be shares in trade: A gains $126,
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 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. same 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 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.
RÚLE.-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. | long would they be in ringing,
3. How many changes may 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.
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 i=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+ (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+. to=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+=3, 0.3=5, &c.
Again, as 0.010101, or 0.0i, is the developement of the 0.02=;g, and so on, and in like manner of 5$, &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.