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

3
It follows that each of the equal parts in - is 4 times as great aeach of the equal parts in -;
and since 3, the same number of parts, is taken in each fraction,,
therefore s is 4 times -2,
or 3 multiplied by 4 gives 3.
5. PROP. A fraction is divided by any number by mullztipying the denoeninator by the number and retaining the nmierator.
Thus 3 divided by 4 gives,5;
for the unit in - is divided into 5 equal parts,
and the unit in,3 is divided into 20 equal parts.
It follows that each of the equal parts in,- is 4 of each of tile
equal parts in -;
and since 3, the same number of parts, is taken in each case,
therefore -3 is - of a,
or that - divided by 4 gives -2.
6. PROP. A fraction is divided by any nm6ber by dividing the nume)raztor
by the number and retaining the dezominator.
Thus 1 - divided by 7 gives -Ts;
for the unit in each of the fractions - and - is divided into 15
equal parts,
and 14 of these parts are taken in one case and 2 in the other;
and since 2 parts are one seventh of 14 parts,
it follows that -~ is one seventh of 4-,
or that - 4 divided by 7 gives 2.
7. PROP. The numerator and denomnzinator of a fraction nmay be muZltillied
by the same nmeber without altering the value of the fraction.l
For if the numerator of any fraction 3 be multiplied by any number 4,
the fraction is multiplied by that number and becomes 1-;
and if the denominator of -5-2 be multiplied by the same number 4,
the fraction is divided by that number and becomes - -.
Now if a quantity be both multiplied and divided by the same number,
its value is not altered.
It follows, therefore, that if both the numerator and denominator
of a fraction be multiplied by the same number, the value of the
fraction is not altered.
91
As any complex fraction as - means no more than 21 is divided by 31, tlhe
reduction of such fractions rather falls under division of fractions. But every complex fraction may be reduced to a single fraction by multiplying the numerator and
denominator by the least common multiple of the denominators, which occur in the
numerator and denominator of the complex fraction: thus, 3~ becomes I.9, by
multiplying the numerator and denominator by 12.