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weighing 43750lbs. gross, allowing 12 per cent. tare, and 7lbs. per hogshead for draft or scalage, at .03 cents per lb.?

Ans. $1150.38.

SINGLE POSITION. SINGLE POSITION is the working of one supposed number, as if it were the true one, to find the true number.

RULE.

1. Take any number, and perform the same operations
with it as are described in the question; then,
2. As the result of the operation

Is to the given number,
So is the assumed number
To the number required.

PROOF.

Perform on the number found, the operation described in the question, and the result will be the given number,

EXAMPLES.

1. Two men, A and B, having found a sum of money, disputed who should have it: A said the half, third, and one-fourth of the money made 130 dollars, and if B could tell how much was in it, he should have it all, otherwise he should have nothing—I demand how much was in the bag. Suppose 60 dollars

As 65 : 130 :: 60

60

= 30

20
15

65)7800(120 dols. Ans.

65

65 result

130 130

2

2. A, B, and C, talking of their ages, B said his age was os

once and half the age of A; and C said his age was twice

and one-tenth the age of both, and that the sum of their Mages was 93—what was the age of each?

Ans. A's 12, B's 18, and C's 63 years. 3. Seven-eighths of a certain number exceeds four-fifths 80 by 6-what is thật number?

Ans. 80.

158

270 DOUBLE POSITION.
4. A's age is double that of B, and B's is triple that of
C's, and the sum of their ages is 140-what is the age of
each?

Ans. A's 84, B’s 42, and C's 14 years.
5. A man carrying a purse of money in his hand, an-
other asked him how much was in it; he answered he could

not tell ; but the third, fourth, and fifth of it made 94 dol47 lars--how much was in the purse ?

Ans. $120. 6. A person lent his friend an unknown sum of money,

to receive interest for the same at 6 per cent. per annum,
540
748 od

simple interest, and at the end of twelve years received for
principal and interest 860 dollars—what was the sum -lent?
184800: $11:1860 ons 10.

Ans. Shyy. 7. A person bought a chaise, horse, and harness, for 100 só dollars; the horse came to twice the price of the harness, 300 and the chaise to twice the price of the horse and harnesswhat did he give for each?

Ans. $11.114 for the harness, $22.224 for the horse,

and $66.66% for the chaise.

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DOUBLE POSITION.
DOUBLE POSITION is used for solving such questions as
require two supposed numbers.

RIJLE.

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Suppose two numbers, and work with each according to the conditions of the question.

Mark the errors with the signs plus and minus, that is to say, when the result proves that the supposed number exceeds the required one, mark the error plus (+); and when the result proves the supposed number to be less than the required one, mark the error minus (-): then place the less error under the larger, and the first supposed number opposite the error of the second operation, and the second supposed number opposite the error of the first operation, with the sign (*) between them; then multiply accordingly. If the errors be both plus, or both minus, divide the difference of the products arising from the errors being multiplied by the supposed numbers, by the difference of the errors : but if one be plus and the other minùs, divide the sum of the said products by the sum of the errors; the quotient, in either case, will be the true nmmber or answer.

EXAMPLES.

1. Divide $1000 so that B may have twice as much as A wanting $80, and C three times as much wanting $150,the share of each is required.

Suppose $100 A 100 X 2480 =120 B 100 x 3-150=150 C

1000-370=630 error

Suppose $200 A
200X280 =320 B
200 x 3-150=450 C
1000-970= 30 error-

600 diff. of errors
200X 630=126000 product
100 x 30= 3000 do.
6,00)123,000 diff. of products

205 A's share
205 X2-80 =330 B's share
205x3-150=465 C's share

Proof 1000 whole sum 2. A and B laid out equal sums of money in trade; A gained a sum equal to $ of his stock, and B lost 225 dollars; then A's money was double that of B's—what did each one lay out?

Ans. $600. 3. A farmer having driven his cattle to market, received for them all $320, being paid at the rate of $24 per ox, $16 per cow, and $6 per calf—there were as many oxen as cows, and four times as many calves as cows—how many were there of each sort? Ans. 5 oxen, 5.cows, and 20 calves.

4. What number is that, which being increased by its }, its 4, and 5 more, will be double ?

Ans. 20. 5. A gentleman going into a garden, met a number of ladies, and said to them, “Good morning to you ten ladies.” “Sir, you mistake," answered one of them, “we are not ten, but if we were thrice as many as we are, we should be as many above ten, as we are now under”-how many

ladis were there?

An

100

160

SHIPS' TONNAGE. 6. There is a fish, whose head is 10 feet long, his tail is as long as his head and half the length of his body, and his body as long as the head and tail-what is the whole length of the fish ?

Ans. 80 feet. 7. Three men, A, B, and C, playing at cards, staked $324; but they happened to disagree ;-each seized as inany of the dollars as he could ; A got a number unknown, B as many as A and 15 over, C got a fifth part of their sums added together-how many dollars did each get? 100

Ans. A $127, B $142, and C $54.

3 45 持将

05*%8:43

726117106 124

SHIPS' TONNAGE. 142을 By carpenters' measyre,

497156-3358 100786

23 166151 - 2300 RULE.

720, 164,5, For sing paddecked vessels, multiply the length, breadth at the main beam, and depth of the hold together, and divide the product by 95; the quotient will be the tonnage required.

For double-decked vessels, take half the breadth of the main beam for the depth of the hold, and work for singledecked vessels.

NOTE.-95 may be considered an arbitrary number. In England 94 is used.

EXAMPLES.

1. I demand the tonnage of a single-decked vessel, mea. suring as follows, viz. length 60 feet, breadth 20 feet, and the depth of the hold 8 feet.

Ans. 1010 tons, or 101.0526 tons. Note. This is the usual method of tonnaging a single-decked vessel, having her deck bolted to the wale. But if it be required that the deck be bolted at any height above the wale, the custom is to pay the carpenter for one half of the additional height, (to which the deck may be thus raised) that is, one half of the differen e being added to the former depth, gives the depth to be used in calculatiug the tonnage.

2. Required the tonnage of a double-decked vessel whose length is 80 feet, breadth 26 feet.

Ans. 284%tons, or 284.631 + 3. What is the tonnage of a double-decked vessel, whose length is 65 feet, and breadth 21 feet 6 inches; and what will it amount to at 16 dollars per ton?

Ans. 158.138 tons; and $2530.2115.

The weight of a ship's burthen is the difference between the bodies of water displaced by her, when in ballast, and when laden.

To ascertain a ship's legal tonnage (by government.)

If the ship be double-decked, take the length thereof from the fore part of the main stem to the after part of the stern-post, above the upper deck; the breadth thereof at the broadest part above the main wales, half of which breadth shall be accounted the depth of the vessel ; then deduct from the length three-fifths of the breadth, multiply the remainder by the breadth, and the product by the depth, divide by 95, the quotient is deemed the tonnage of such vessel.

If the ship be single-decked, take the length and breadth as above directed, deduct from the length, three-fifths of the breadth, take the depth from the under side of the deck plank to the ceiling in the hold, and multiply and divide as aforesaid for the tonnage.

(Laws of United States, chapter 128, sec. 64.) Though this act does not indicate by what specific measures this length, breadth, and depth are to be taken, yet it has always been understood here to be in feet and tenths of a foot. [Vide Adams' Report on Weights and Measures, p. 65.]

EXAMPLES.

1. Required the legal tonnage of a double-decked vessel, the length being 110.5 feet, and the breadth 30.6 feet?

of 30.6=18.36, this subtracted from the length, thus 110.5-18.36=92.14

2)30.6
30.6 breadth

15.3 depth
55284
27642

2819.484

* 15.3 depth

8458452 42292260

95)43138.1052(454.085 tons.

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