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United States tonage is thus found :

Rule. Take the length from the fore part of the stem, to the hind part of the stern post, above the upper decks, the breadth at the broadest part above the main wales; half this breadth is called the depth ; then deduct from the length of the breadth, the remainder multiply by the breadth, and that product by the depth, and divide this last product by 95, for tons...

For single decked vessels.

RULE. Take the length and breath as above, and subtract of the breadth from the length the remain. der multiply by the breadth, and that product by the depth from the under side of the deck plank, to the ceiling at bottom; divide this last product by 94 for tons.

EXAMPLES 1. What is the tonage of a ship, 87 feet from the fore

part of the stem, to the hind part of the stern post above her upper deck, and 25 feet wide, United States tonage ?

Ans. 236 % 2. What is the tonage of a schooner (single deck) 69 feet from the fore part of the stem, to the hind part of the stern post, 17 feet wide, and 7 feet deep ?

Ans. 7932

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Vulgar Fractions are parts of an unit, (or whole number) and are expressed by two numbers with a line drawn between them, the number over the line is called the numerator, and that under it the denominator.

The denominator is the number of parts the unit is supposed to be divided into, and the numerator is the number of those parts the fraction contains; į is one half, i. e. an unit divided into halves, or two parts, this is one half, or one of those parts; is three fourths,

is twenty five twenty sixthis, i. e. if an unit be divided into twenty six parts, it is twenty five of those parts.

25

3 4 15 2: 379

1$, &c.

Note. A remainder after division is a numerator of a fraction, of which the divisor is its denominator.

Vulgar Fractions are of four kinds, viz. pro per, improper, compound, and mixed.

A fraction is said to be proper when the numerator is less than the denominator ; thus, in

Improper fractions have their numerators greater than their denominators ; thus, is, 1 , &c.

A compound fraction is a fraction of a fraction, thus,

of j of 1 of 2, &c. i. e. one half of three fourths, two thirds of one half, three eighths of nine twelfths, &c.

A mixed fraction consists of an integer, or whole number, with a fraction annexed ; thus 7, 127, are expressed seven and three ninths, twelve and seven eighths, &c.

NOTE. Any whole number may be reduced to an inproper fraction, by considering it as a numerator; and placing 1. under it for a denominator, with a line drawn between then, thus, i 147 i. e. eight ones, fifteen ones, one hundred and twenty seven ones ; here the 8, the 15, and the 127, are reduced to improper fractions.

REDUCTION OF VULGAR FRACTIONS.

CASE I. To reduce vulgar fractions to their lowest terms,*

RULE. Divide the numerator and denominator by such a number as will divide them both without a remainder; then divide these quotients (if need be) by the same, or any other number which will divide both without a remainder, &c. till no number except I will divide so : Or, find a common measure for dividing them at once ; thus,

Divide the denominator by the numerator, then divide the numerator by the remainder, and then divide

36

48

* Fractions are said to be in their lowest terms when they are expressed by the least numbers possible, thus,

12969 576) &c. when reduced to their lowest terms, will be a

&c.

369 123

1

the first remainder by the second, and the second remainder by the third, &.c. till nothing remains; the last divisor will be the common measure by which the numerator and denominator must be divided, their quotients will be the lowest terms possible to reduce the fractions to.

EXAMPLES. 1. Reduce is to its 2. Reduce me to its lowest lowest terms.

876(4 Ans. First method. 121813105( 35 Ans.

3. Reduce yi to its low

est terins.
Second method.
36)1296(36

4. Reduce 4*, to its low108

est terms.

terms.

Ans. 4

Ans

..

12

4 6 8 11879

to its Ans. 117

216

5. Reduce 60m, to its low

12T
216
est terms.

Ans. As nothing remains, 36 the divisor is the common 6. Reduce 153 to its lowmeasure.

est terms,

Ans. 1 36)1296(36 new de108 nominator. 7. Reduce lowest terms.

. . 216 216

8. Reduce

5 96 1275

to it lowfest terms.

1275 36)36(1 new numera.

36 tor, therefore the fraction in its lowest terms is 36

CASE II. To reduce vulgar fractions of different denominations, to a common denominator.

RULE. Multiply each numerator into all the denom. inators, but its own, for a new numerator.

2. Multiply all the denominators together, for a common denominator.

Ans. -596 *

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As no number, except 1, will divide these, without a remainder, it is already in the lowest terms possible.

EXAMPLES.
1. Reduce ; and, to a common denominator.

an

Ans. and 2. Reduce }, Tó, and 11, to a common denominator.

5 6 30

8 6 48

12

7 10 70 12

11

8 88

10 887

727

8

100

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4320 500

Ans. ** $6 and 466 3. Reduce, ta, , and į, to a common denominator,

Ans. 07. 111, 1:1, and Hie 4. Reduce 18, 9, 3, and 4, to a coinmon denominator.

Ans. 3024, 25 20 5. Reduce , 1,, and , to a common denominator.

Ans. 384, 240, 400, 120 If these numerators were added together, and their sum exceed the denominator, then if the denominator be filaced under the sum with a line drawn between them, it would be an improper fraction. See the result.

187* Improper fraction.

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384 2.40 400 120 1144

CASE III. To reduce a mixed number to an im

proper fraction.

RULE. Multiply the whole number by the denominator of the fraction, and take in, or add to the product the numerator, their sum will be a new bumerator, under which place the denominator of the fraction, with a line drawn between them, and they will be the improper fraction required.

EXAMPLES
1. Reduce 12 iš to an improper fraction.

12

99 12

Ans.

219

2. Reduce 1678, to an improper fraction.

16700

Ans. 16 18

100.

1618 3. Reduce 1945 to an improper fraction.

Ans. 13. 4. Reduce 16g, to an improper fraction.

Ans. 5. Reduce 19ş, to an improper fraction..' Ans..

CASE IV. To reduce an improper fraction to its proper terms.

Rule. Divide the numerator by the denominator.
Nore. This case and Case III. prove each other.

EXAMPLES.
1. Reduce 10, to its proper terms.

17)219(12+ Ans.

17

49 34

3 5 4

to its

T8

1817

15
2. Reduce
16-18, to its proper terms.

Ans. 16106
10
3. Reduce

proper terms.

Ans. 1915. 4. Reduce

Ans. 1634
TT2)

to its proper terms.
5. Reduce mais
to its proper terms,

Ans. 19
CASE V. To reduce compound to simple fractions.

Rule 1. Multiply all the numerators together for a new numerator.

2. Multiply all the denominations together for a new denominator.

EXAMPLES.
1. Reduce 4 of of to a single fraction.

2

3

Ans. 22=

3
7 new aumerator.

7 new denominator.

21

2. Reduce of of is to a single fraction.

Ans. = 3. Reduce 14 of of g to a single fraction.

Ans. 1883

13

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