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

7
Iind the least common multiple of all the denominators.
Divide this number by each of the denominators respectively,
and then multiply the numerator and denominator of each of the
fractions given by its corresponding quotient.1
An integer may be reduced to a fraction with any given denonminator by writing the integer as a fraction with unity for its denominator,
and then multiplying the numerator and denominator by the given
denominator.
13. PROP. To find the sztm of two or more fractions.2
If the fractions have the same common denominator, their sum
will be expressed by the sum of the numerators taken for a new
numerator and placed over the common denominator.
When the fractions have not the same denominator, when reduced
to their least common denominator their sum can be found.
In case of mixed numbers, in practice, it will be found most convenient first to find the sum of the fractional, next of the integral
parts, and then the sum of both will be the sum required.
Instead of adding all the fractions together by one operation, it
will sometimes be found convenient first to find the sum of two of the
fractions, next to add a third fraction to this sum, and so on until all
the fractions have been added.
14. PROP. o find the difference of two fractions.
If the fractions have a common denominator, their difference will
be expressed by the difference of the numerators placed over the
common denominator.
If the fractions have different denominators, they must be reduced
to their least common denominator, and then their difference can be
found.
Instead of reducing mixed numbers to improper fractions, and
then finding their difference, it will be found more convenient in
practice to reduce the fractional parts only to a common denominator,
and to find first the difference of the fractional parts, next of the
integral parts.3
' Two or more fractions may be compared, or their relative values ascertained,
by reducing them to the same common denominator.
It may be shown, in general, that if any number be added to, or subtracted from,
the numerator and denominator of a fraction, the value of the fraction is either ilcreased or decreased in value.
2 In the operations of addition and subtraction of integers, the units must all be
[of the same magnitude, in order that the sum and difference of any given numbers
mlay be expressed by one number: so also in fractions, the parts of the unit expressed
by each fraction must be of the same magnitude before the sum or difference of any
fractions can be expressed as one fraction. If each of the given fractions express
different parts of the unit, they must be converted to others of equivalent values,
expressing the same parts of the unit; or in other words, be reduced to the same
common denominator.
3 If the numerator of the fractional part to be subtracted be greater than that