18sh. 4d; and the addition is performed as in the margin,

farthings, equal to 2d, which being carried to the pence column, the addition is performed in the usual way.

SUBTRACTION OF VULGAR FRACTIONS.

Subtraction is performed by reducing the two given frac tions, if necessary, to the same denomination, and then, subtracting the less numerator from the greater, for a new numerator to the common denominator: thus if from we take away equal to 2, it is evident that will remain: and in the same way to subtract from 35, we first reduce them to fractions of a common denominator, viz. to 56, and to; then subtracting the less numerator 265, from the greater 792, we have 527, for the numerator of the remainder 5277

255

In subtracting mixed numbers, supposing the fractional parts to be either given in or reduced to the same denomin ation, if the fractional part of the least quantity be greater than that of the greater quantity, then an unit must be borrowed, containing a number of parts equal to those in the common denominator, and added to the less fraction; and this unit must be repaid to the unit of the subtrahend, as in subtraction of integers.

In the annexed case the two fractions must be reduced to the same denomination; but as 49 cannot be taken from, we borrow an

581 or

男

37% or

2014or

unit from the integral part of the fraction, and dividing it into the same number of parts with the common denomi

nator

ator, viz 48, we add 48 to 18, making 66, from which subtracting the numerator of the other fraction 40, we have a remainder of 3, or when reduced to its lowest common measure, 4 be written in the remainder, and carrying the unit borrowed to the integer 7, the rest of the subtraction is performed as in common arithmetic.

13

nomination with the 3d of the Subtrahend, we must borrow an unit or that is 4 farthings from the pence, making with the ord. d, and from this subtracting the given we have a remainder of to be written down, and d or ld to be carried to the column of pence; the rest of the subtraction proceeding as formerly shown.

MULTIPLICATION OF VULGAR FRACTIONS.

This operation consists in multiplying the numerators of the two given fractions into one another for a new numerafor, and the denominators together for a new denominator to the product: thas in multiplying by, the product of the numerators being 6, and that of the denominators being 12, we have the fractional product, or, for the result. This will be easily understood by taking of a foot to be multiplied by of a foot : for 3 being 8 inches, and being 9 inches, these two quantities multiplied together will give 72 square inches: but it was formerly mentioned that the square inches in one foot are 14, that is 12 times 12; 72 inches are therefore of 144, or of 1 foot, agreeing to, being the product of 3 by 2.

Hence it is to be remembered that fractions multiplied together give products of less numerical value than the factors, and that even units multiplied by fractions give products of less apparent value than the units: for 1 foot multiplied by

foot,

foot, that is 12 inches multiplied by 9 inches will give foot square or 108 inches: but this arises from the difference between the lineal measure of the factors, and the superficial measure of the product, which however are both expressed by the same terms and figures.

When integers are multiplied by integers, their value is increased; when they are multiplied by units or 1, their value remains unaltered; but when multiplied by fractions their value is diminished proportionally to the difference be tween the fraction and unity.

To multiply an integer by a fraction, you multiply the integer by the numerator and divide the product by the de

846 by

5

8)4230(

nominator, and if there be any remainder it is annexed to the quotient as the numerator of a fraction with the same denominator 52832 as that in the multiplier. See the example. This however is only an application of the rule already given, for the integer here may be represented by an improper fraction whose numerator is the given integer, and whose denominator is 1: thus in the example, the whole number 846 may be fractionally expressed as 846.

To multiply an integer by a mixed number, first multiply it by the integral part, and to the product add that obtained by multiplying the integer with the fractional part. For instance let it be required to multiply 365 by 64.

Multiply the given

fractional part, and dividing the product by the denomininator 5, add the quotient to the first product, and the sum will be the product required.

In multiplying a mixed number by a fraction, first multiply the fraction into the integral parts, and then into the fractional; and the sum of these products will be the product required. For example multiply 35 3 by

Lastly, in multiplying two mixed numbers together, add the products of the two integral and the two fractional parts alternately multiplied into one another, and their sum will be the product required. Multiply, for example, 183 by 121.

To illustrate this example: suppose the quantities to be multiplied are 183 feet, or 18 feet 8 inches, and 12 feet, or 12 feet 9 inches: then by the rule given when treating of multiplication of compound quantities, the process will be

as here shown, where the product of 18f. 8in. by 12f. 9in.

is 238 feet, as was found in the operation by multiplication of mixed numbers.

DIVISION OF VULGAR FRACTIONS.

Division is performed by multiplying the numerator of the dividend by the denominator of the divisor, for a numerator to the quotient, and the denominator of the dividend by the numerator of the divisor, for a denominator to the quotient. For example, let it be required to divide ♣ by, we multiply the numerator of the dividend 3, by the denominator of the divisor 3, and the product 9 is the numerator of the quotient: again the numerator of the divisor 2 multiplied into the denominator of the dividend 4 gives 8 for the denominator of the quotient, which then becomes the improper fraction equal to 1.

In this operation the object being to discover how often are contained in 2, it is evident that two thirds will be contained in any quantity only half the number of times that one third would be contained in it: we therefore divide the dividend by 2 and multiply the quotient by 3, or in other words, we take 3 times the half of the dividend; for the quotient required by the question proposed. Thus in the example given; bring the given dividend to some equivalent fraction that will admit of division by 2, as ģ; the half of this is and 3 times the quotient is equal to 1, as before found. This will be illustrated if we suppose the fractions given to be parts of a foot for instance, in which the divisor will be 8 inches and the dividend will be 9 inches and here it is evident that 8 will be contained in 9, once with one over; or the quotient will be 1 as before.

In dividing an integer by a fraction, or a fraction by an integer, you must bring the integer into a fractional form by writing 1 under it, for a denominator, and then working as in the preceding example: thus to divide 5 by the operation would be performed as in the margin, where the

quotient