IMPROPER FRACTIONS
Although the "improper" fraction is really quite
"proper" mathematically, it is usually customary
to change it to a mixed number. A recipe may
call for 1 1/2 cups of milk, but would not
call for 3/2 cups of milk.
Since a fraction is an indicated division, a method
is already known for reduction of improper fractions to mixed numbers. The
improper fraction 8/3 may be considered as the division of 8. by 3. This
division is carried out as follows:
The truth of this can be verified another way:
If 1 equals 3/3 then 2 equals 6/3 Thus,
These examples lead to the following conclusion, which is stated as a rule:
To change an improper fraction to a mixed
number, divide the numerator by the
denominator and write the fractional part of
the quotient in lowest terms.
Practice problems. Change the following fractions
to mixed numbers:
1. 31/20
2. 33/9
3. 65/20
4. 45/8
Answers :
OPERATING WITH MIXED NUMBERS
In computation, mixed numbers are often unwieldy. As it is possible to change
any improper fraction to a mixed number, it is like wise
possible to change any mixed number to an improper
fraction. The problem can be reduced to the
finding of an equivalent fraction and a simple
addition.
EXAMPLE: Change to an improper fraction.
SOLUTION:
Step 1: Write as a whole number plus a
fraction, 2 +1/5.
Step 2: Change 2 to an equivalent fraction with
a denominator of 5, as follows:
Step 3: Add
Thus,
EXAMPLE: Write as an improper fraction.
SOLUTION:
Thus,
In each of these examples, notice that the multiplier
used in step 2 is the same number as the
denominator of the fractional part of the original
mixed number. This leads to the following conclusion, which is stated as a rule:
To change a mixed number to an improper fraction,
multiply the wholenumber part by the denominator
of the fractional part and add the numerator
to this product. The result is the numerator
of the improper fraction; its denominator is the same as the denominator of the
fractional part of the original mixed number.
Practice problems. Change the following mixed
numbers to improper fractions:
Answers:
NEGATIVE FRACTIONS
A fraction preceded by a minus sign is negative. Any negative fraction is
equivalent to a positive fraction multiplied
by 1. For example,
The number 2/5 is read "minus twofifths."
We know that the quotient of two numbers with
unlike signs is negative. Therefore,
This indicates that a negative fraction is equivalent to a fraction with
either a negative numerator or a negative denominator. The
fraction 2/5 is read "two over minus five."
The fraction 3 is read "minus two over
five."
A minus sign in a fraction can be moved about
at will. It can be placed before the numerator, before the denominator, or
before the fraction itself. Thus,
Moving the minus sign from numerator to denominator,
or vice versa, is equivalent to multiplying
the terms of the fraction by 1. This is
shown in the following examples: A fraction
may be regarded as having three signs
associated with itthe sign of the numerator, the sign of the denominator, and
the sign preceding the fraction. Any two of
these signs may be
changed without changing the value of the
fraction. Thus,
OPERATIONS WITH FRACTIONS
It will be recalled from the discussion of denominate
numbers that numbers must be of the same
denomination to be added. We can add pounds
to pounds, pints to pints, but not ounces to
pints. If we think of fractions loosely as denominate numbers, it will be seen
that the rule of likeness applies also to
fractions. We can add eighths to eighths,
fourths to fourths, but not eighths to
fourths. To add 1/5 inch to 2/5 inch we
simply add the numerators and retain the denominator
unchanged. The denomination is fifths; as
with denominate numbers, we add 1 fifth to 2
fifths to get 3 fifths, or 3/5.
