3 What sum is that, of which the ‡, and make 74. ansiver 1201.

4 What sum of money, at 6 per cent. per annum, simple interest, will amount to 500l. in 10 years ? answer 3121 10s.

5 Three unequal vents will severally empty a vessel of 120 gallons in I hour, 2 hours, and 3 hours; if running together, what time is necessary? answer 32min. 437 sec. 6 Of a certain sum given A, B1, B1, and D the rest, which is 28. the sum is required?

answer 112/. 7 What is the age of a person who says, that if of the years I have lived be multiplied by 7, and ed to the product, the sum will be 292? 8 Required the sum, the 4, 4, and of

9 What sum, at 6 per cent. per annum, 8601. in 12 years ?

៖

of them be addanswer 60 years. which made 941.

answer 1201. will amount to

answer 5001. 10 A person having about him a certain number of dollars, said, that, 1, and of them would make 57; how many had he? answer 60.

11. A schoolmaster being asked how many scholars he had, answered, if to double the number I add,, and 4 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 50l. 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 of the principal by an 100l. and she does not intend to marry any man, who is not scholar enough to tell her fortune; pray what is it ? answer 10000l.

DOUBLE POSITION.

Double position teaches to solve such questions as require 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 number of the other; then,

If the errors be alike, i. e. both too much, or too little, take their difference for a divisor, and the difference of the product for a dividend: but if unlike, take their sum for a divisor, 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.

EXAMPLES.

5

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, 2d. 150

2 Divide 100l. so that B may have twice as much as A, wanting 81. and C three times as much, wanting 157. what is each man's share?

answer A 20l 10s. B 381. C 46l 10s. 3 Of 100l. expenditures, B paid 10l. 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 18 8d. how many days did he work, and how many was he idle? answer wrought 30 days, idle 10. 7 Bought 15 yards for 3 10s. viz. damask at 8s. per yard, and lining for it, at 3s. per yard; what quantity was

there of each ?

answer

S5 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 P answer 6001.

9 When first the marriage knot was ty'd
Between my wife and me,

My age was to that of my bride

As three times three to three;
But now when ten, and half ten years,
We man and wife have been,

Her age to mine exactly bears,
As eight is to sixteen:

Now tell, I pray, from what I've said,
What were our ages when we wed?

answer

Thy age when marry'd must have been
Just forty-five; thy wife's fifteen.

PERMUTATION.

P

PERMUTATION.

ERMUTATION is a rule for finding how many different ways any given number of things may be varied in positions, or succession; thus, abc, acb, bac, bca, cab, cba, are six different positions of three letters.

RULE.

Multiply all the terms of the natural series continually from 1 to the given number inclusive, the last product will be the changes required.

EXAMPLES.

1. In how many different positions can 5 persons place themselves at a table?

1X2X3X4×5=120 answer.

2 What number of changes may | zbe rung upon 12 bells, and in what time may they be rung, allowing 3 seconds to every round? S479001600 changes. answer 45 years, 195 days, 18 hours. 3 What time will it require for 8 persons to seat themselves every day differently at dinner? ans. 110gr. 142da. 4 What number of variations will the 26 letters of the alphabet admit of P ans. 403291461126605635584000000

C

COMBINATION.

OMBINATION 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, ác, bc.

RULE.

Take a series proceeding from and increasing by a unit, 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.

EXAMPLES.

1 How many combinations of 5 letters in 10?