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3. What sum is that, of which the , and ; make 74.
answer 1201. 4 What sum of money, at 6 per cent. per annum simple interest will amount to gool. in ro years ? answer 3121 10s.
5 Three unequal vents will severally empty a vessel of 120 gallons in 1 hour, 2 hours, and 3 hours ; if running together, what time is necessary ?
answer 32 min. 43 ' secó 6 Of a certain sum given A }, B, C, and D the rest, which is 281. the sum is required ?
answer 1121. 7 What is the age of a person who says, that if of the years I have lived be multiplied by 7, and of them be added to the product, the sum will be 292 ? answer 60 years. 8 Required the sum, the }, \, and, of which made 94).
answer Izol. 9 What sum, at 6 per cent. per annum, will amount to 8601. in 12 years ?
answer 5ool. IC A person having about him a certain number of dollars, said, that }, ,
many had he?
answer 60. 11 A schoolmaster being asked how many scholars he had, answered, if to double the number I add * and of them, I shall have 333 ; how many had he? answer 108
12 A saves of his income; but B, who has the same salary, by living twice as fast as A, sinks gol. a year ; how much then have they per annum? answer 150l. each.
13 The yearly interest of Charlotte's money, at 6 per cent. exceeds zo of the principal by an 100l. and she does not intend to arry any man, who is not scholar enough to tell her fortune ; pray what is it;
Double position teaches to solve such questions as re. quire two supposed numbers in the operation.
RULE. Suppose two numbers, and work with each agreeably to the tenor of the question, noting the errors of the results ; multiply the errors of each operation into the supposed num. ber of the other; then,
If the errors be alike, i.e. both too much, or too little, make their difference for a divisor, and the difference of the
pro. duct for a dividend : but if unlike, take their sum for a divi. sor, and the sum of the products for a dividend. Note. In many instances, if o be used for the first, and 1 for the second
of the supposed number, the first of the errors, divided by their difference will be the answer. Proof as in single position.
E x A M P L E s. 1 A farmer hired a labourer on this condition, that for every day he worked, he should receive 12d. but for every day he was idle he should be fined 8d. when 390 days were past, neither of them was indebted to the other ; how many days did he work.
Suppose ist. 140 working days,
240 140 X 12= 1680 earned
150X12=1800 250 X 8 2000 fined
240 X 8=1920
2 Divide 100l. so that B may have twice as much as A, wanting 8l. and C three times as much, wanting 15l. what Is each man's share?
answer A 204. 10s. B 331. C 461 10s. 3 Of 1001. expenditures, B paid !07. more than A, and C as much as A and B ; each man's part is required ?
answer A 201. B 301. C 501. 4 A is 20 years of age : B's age is A's and half C's; and C's equals them both; their several ages are required ?
answer A 20, B-60, C 80 years. 5 The head of a fish is 9 inches long, and its tail is as long as its head and half the body, and the length of the body equal those of the head and tail ; what is its whole length ?
answer 6 feet. 6 A labourer hired for 40 days upon this condition, that he should receive 20d. for every day he wrought, and forfeit 10d. for every day he was idle ; at settlement he received 21 Is 8d. how many days did he work, and how many was he idle ?
answer wrought 30 days, idle 10. 7 Bought 15 yards for 31 10s. viz. damask at 8s. per yard, and lining for it, at 3s. per yard; what quantity was there of each?
5 yards damask,
10 ditto lining 8 A and B put equal sums of money in trade ; A gained a sum equal to of his stock, and B lost 2251. then A's money was double that of B's; what capital did each of them begin with
answer 6001. 9 When first the marriage koot was ty'd
Between my wife and me,
As three times three to three ;
We man and wife have been,
As eight is to sixteen :
when we wed ?
PERMUTATION. ERMUTATION is a rule for finding how many
different ways any given number of things may be varied in position, or succession ; thus, abc, acb, bac, bca, cab, cba, are six different positions of three letters.
RUL E. Multiply all the terms of the natural series continually from 1 to the given number inclusive, the last product will be the changes required.
EXAMPL E S. i In how many different positions can 5 persons place themselves at a table ? 1 X2 X3 X4 X5=120 answer.
2 What number of changes may be rung upon 12 bells, and in what time may they be rung, allowing 3 seconds to
$ 479001600 changes.
145 years, 195 days, 18 hours. 3 What time will it require for 8 persons to seat themselves every day differently at dinner? ans. 110yr. 142 days.
4 What number of variations will the 26 letters of the al. phabet admit of ? ans. 403291461126605635584000000
COMBINATION. NOMBINATION discovers how many different ways
a less number of things may be combined out of a greater ; thus, out of the letters a, b, c, are three different combinations of two, viz. ab, ac, bc.
RULE. Take a series proceeding from and increasing by a unity up to the number to be combined ; and another series of as many places, decreasing by unity, from the number out of which the combinations are to be made; multiply the first continually for a divisor, and the latter for a dividend, the quotient will be the answer.
ΕΧ Α Μ Ρ ι ι ς.
252 answer. IX X3 X4x8
2 What is the value of as many different dozens as may be chosen out of 24 at id. per dozen? ans. 112671 6s 4d. 3 How
many different ways may a butcher select 50 sheep out of a fock consisting of 100, so as not to make the same choice twice?.. ans. 10891306544874079257172497236
The denominations are :
I Inch 1. 12 Inches
1 Foot Fi. ADDITION OF DUODECIMALS
E x A M P LE S.
28 4 3 y lo
71 7 8 4 2 56 10 5 7 9
67 Il 36 43 1 6 4 3
32 0 8 4 con 87 11 10 8 5
46 3 8 11 10 48 5 2 10 11
67 11 9 4 11
336 5 I 4
1 Five floors in a certain building contain each 1295 gi. 8" how many feet in all ? answer 6479f. oi. 4".
2 Several boards measure as follow: viz. 274. zi. 254. 11. 23f. 1oi. 2of. i. 20f, 6i. and 188. si what number of feet do they contain ?
answer 136f. Sie SUBTRACTION OF DUODECIMALS.
RULE. Work as in compound subtraction, borrowing 12, when necessary.