Elementary arithmetic, with brief notices of its history... by Robert Potts.

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15. PROP. o find tle product of two or more fractions.1
DEF. To multiply one fraction by another, is to take such part or
parts of one as are expressed by the other; and this is effected by
dividing the former fraction by the denominator, and multiplying it by
the numerator of the latter fraction.
To find the product of- by 4.
Here the multiplication of ] by 4 implies
that 3 is to be divided by 7, and the result multiplied by 4.
Now 3 divided by 7 gives 3 7;
and this result, _3, multiplied by 4, produces 3X4
and, therefore, 3 x 4-= 4, or.
Hence the product of two fractions is found by multiplying the
two numerators for the numerator, and the denominators for the denominator of the product.
In the same way it may be shewn that the product of more fractions than two is found by multiplying all the numerators together,
and all the denominators together, for the numerator and denominator
of the product.
16. PROP. To divide one fraction by another.2
from which it is to be taken, the subtraction is not possible. But the subtraction
can be made possible by increasing the latter by a fraction equal to unity, and the
other by unity. Thus in finding the difference of 153 and 8; 4 cannot be taken
from 4, but if 7- be added to 156, and 1 to 84, the subtraction then becomes possible,
as " the difference of two numbers is not altered when each of the numbers is equally
increased."
1 In the multiplication and division of fractions, it will generally be found most
convenient, first to reduce mixed numbers and compound fractions to simple
fractions.
There is an impropriety of language in the employment of the word multiply
in reference to proper fractions. To multiply two proper fractions is to take of
neither of them so much as once, but only that part of one fraction which is
expressed by the other: since a proper fraction is always less than unity, it
follows that the product of two proper fractions will be less than either of them,
and the multiplication produces decrease and not increase of magnitude.
In practice, however, it will be found more convenient in the multiplication of
fractions to divide any factors of the numerator and denominator which are by
the same numbers divisible, and to use the quotients instead of them, and the
product will be expressed in its lowest terms.
1 5 6 9
Thus to multiply-, —, -, 9 together,
e x 5 x 6 x9 1 xl x x 3 _ 3
here-x-x-x3 6 7 20 1 x 1 x 7 x 4 28
First, 3 in denominator and 9 in numerator are divisible by 3; next, 5 in
numerator and 20 in denominator are divisible by 5; thirdly, 6 in denominator and
6 in numerator are divisible by 6, and the respective quotients are used instead of
the given numerators and denominators. If, however, the numerators are multiplied
for the new numerator, and the denominators for the new denominator, the fraction is
2 7 - which, when reduced by 90 the greatest common measure, becomes 2-.
2 The quotient of one fraction divided by another may be found by dividing the