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

4
8 PO. P. The numerator and denominator of a fraction may be divided by
the same number without altering the value of the fraction.1
For if the numerator of a fraction - 2 be divided by any number 4,
the fraction is divided by that number and becomes.j;
and if the denominator of 3 be divided by the same number 4,
the fraction is multiplied by that number and becomes s-.
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
any fraction be divided by the same number, its value is not altered.
Hence it appears that the value of a fraction depends not on the
absolute, but on the relative values of the numerator and denominator;
and that by means of these two principles, fractions may be altered
in form while they retain the same value.
9. PROP. To convert a mixed number, or an integer and a fraction, into
an improper fraction; and conversely.
To convert a mixed number, as 35, to an improper fraction.
The mixed number 3-5 consists of 3 units and A. of a'unit.
Here the unit being supposed to be divided into six equal parts,
the 3 units will be equal to 18 sixths of the unit,
and the 3 units and 5 sixths will be equal to 23 sixths;
that is, 3- or 3- 15 =- 1=+5= XG5. 23
Hence a mixed number may be converted into an improper
fraction by multiplying the integer by the denominator of the fraction and adding the numerator, and retaining the same denominator.
Conversely, to reduce a mixed number into an improper fraction.
To reduce the improper fraction 23 to a mixed number.
Here 3 18 18+ 5=3+t or 35.
Hence if the numerator of an improper fraction be divided by the
denominator, the quotient is the integral part, and the remainder
(if any) is the numerator of the fractional part, the denominator remaining unchanged.
10. PROP. To reduce a compound fraction to a single one.
For if -I of 4 be any compound fraction,
it implies that the fraction 4 is to be divided into three equal parts,
and two of them are to be taken;
or that 5- is to be divided by 3, and the result multiplied by 2.
1 This principle of dividing the numerator and denominator by the same number
may be applied to find the limits within which any given fraction lies.
Thus to show that.1 7 lies between -1 and -I; 257 divided by 17 gives 15;
17 1r
and 15- is greater than 15 but less than 16; therefore 21 or - is greater than
257 15l
-1- but less than 5, and consequently lies between 25 and 1
16T5 -1 and T 3,