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2 Divide 100l. so that B may have twice as much as A, wanting 81. and C three times as much, wanting 15l. what is each man's share? !

answer A 20l 10s. B 33l. 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 20l. В 30l. C 50l. 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 1s 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 ?

1

S-5 yards damask. ansiver 210 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 600l.

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

My age was to that of

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,

PERMI

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? 1×2×3×4X5=120 answer.

answer

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2 What number of changes may be rung upon 12 bells, and in what time may they be rung, allowing 3 seconds to every round? $479001600 changes.. 45 years, 195 days, 18 hours. 3What time will it require for 8 persons to seat themselves every day differently at dinner? ans. 110yr. 142da. alphabet admit of? ans. 403291461126605635584000000

4 What number of variations will the 26 letters of the

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, ac, 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?

2

2 2
×9×8×7×6=252 answer.
1X7X3×4×8

2 What is the value of as many different dozens as may be chosen out of 24, at 1d. per dozen? ans. 112671 6s 4d. 3 How many different ways may a butcher select 50 sheep out of a flock containing 100, so as not to make the same choice twice? ans. 10891306544874079257172497256

D

DUODECIMALS.

UODECIMALS are fractions of a foot, or of an inch, or parts of an inch, having 12 for their denominator. The denominations are;

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Add as in compound addition, carrying one for each 12 to the next denomination.

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1 Five floors in a certain building contain each 1295f. 9i. 8" how many feet in all ?

answer 6479f. Oi. 4". 2 Several boards measure as follow: viz. 27f. 3i. 25ft. 11i. 23f. 10i. 20f 9i. 20f. 6i. and 18f. 51. what number of feet do they contain ? answer 136f. 8i. SUBTRACTION OF DUODECIMALS.

RULE.

Work as in compound subtraction, borrowing 12, when necessary.

P

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Rem. 78 3 0 10 11

2 From a board measuring 41ft. 7in. cut 19ft. 10in. and what is left? answer 21ft. 9in. MULTIPLICATION OF DUODECIMALS.

CASE 1.

When the feet of the Multiplier does not exceed 12;

RULE.

Set the multiplier in such order that the feet thereof may stand under the lowest denomination of the multiplicand, and in multiplying carry one for every 12 from one denomination to another, and place the result of the lowest demination in the multiplicand under its multiplier.

Note 1. If there be no feet in the Multiplier, supply their place with a cipher. 2. Whether wee begin with the highest or lowest denomination of the multiplier, the several denominations of the products will be respectively synonymous with those of the multiplicand under which they are placed.

EXAMPLES.
Ft. I.

167

Ft. I.

Ft. I. "

Multiply by sft. bin.

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Product 27 16

2 A mahogany board measures 28ft. 10in. 6" by 3ft. 2in. 4", what is its content?

answer 92ft. 2in. 10" 6”” (”””

CASE 2.

When the feet of the multiplier exceeds 12;

RULE.

Use the component parts of the feet in the multiplier as in compound multiplication, and take parts for the inches, &c.

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0 0 7 11

Product 11402

2 A partition is 82ft. 6in. by 13ft. 3in. how many square feet does it contain ?

answer 1093f. li. 6" 3 A floor is 79f. 8i. by 38f. 11i. how many square feet are therein ?

4 If a ceiling be 59f. 9i. long, many yards does it contain ?

5 There is a yard of 21,5 feet be paved with stones of 18 inches are necessary for the purpose ?

answer 3100f. 4i. 4" and 24f. 6i. broad; how answer 162 yards 5f.+ by 17,5 feet; which is to how

square;

stones

many answer 167+

6 Suppose the dimensions of a bale to be 7 feet 6 inches, 3 feet 3 inches, and 1 foot 10 inches; what is the solid content?

ft. in.

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= 22 '6

1 10 6

24 4 6

1 10

4 6x 1ft.24

24 4

4 6

6x10in. 20 3 9

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44 8 3

answer 44 feet 8 inches and 3 twelfth parts.

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