Fractions

One
of the leaves in the notebook is missing. The leaf that follows begins
midsentence describing mixed numbers (for example, 4⅔ is a mixed number)
and then moves on to define mixed fractions. These are fractions or mixed
numbers expressed as fractions (for example, 5⅜ / 4⅔). This is
followed by a definition of a compound fraction. These are factions or mixed
numbers connected by a multiplication sign (×) or the word “of” (for example, 3
× ⅞ of 9˝, which is the same as 3 × ⅞ × 9˝).

The
section concludes with some rules of arithmetic for fractions.

1. *Addition or subtraction of fractions*. Add or subtract
the numerators of the fraction when the denominators are the same.

For example, .

2. *Multiplication of a fraction*. Either multiply
the numerator or divide the denominator.

For example, , or .

3. *Division of a fraction*. Either multiply
the denominator or divide the numerator.

For example, or .

4. *To change the terms of a fraction without changing
its value*.
Multiply the numerator and denominator by the same amount.

For example, .

5. *Inverse*. Take any fraction and invert it (i.e. the
numerator becomes the denominator and the denominator becomes the numerator).
Then the product of the two fractions has the value 1. Thus the *reciprocal* of any fraction is 1 divided
by the fraction.

For example, .

is the reciprocal of .

6. *To multiply by a fraction*, divide by its
reciprocal and *to divide by a fraction*
multiply by its reciprocal.

7. For several factions
of the same value, the sum of the numerators divided by the sum of the
denominators will have the same value as the original fractions. For two
fractions of the same value, the difference in the numerators divided by the
difference in the denominators will have the same value as the original
fractions.

For example, for , , and , .