The following important fact is illustrated in the preceding examples.
Mixed number = natural number + proper fraction
Mixed numbers are the
sum of a natural number and a proper fraction. Mixed number = (natural number) + (proper fraction)
For example
can be expressed as
The fraction
can be expressed as
.
It is important to note that a number such as
does
not indicate multiplication. To indicate multiplication, we would need to use a multiplication symbol (such as ⋅)
means
and
not
, which means 5 times
or 5 multiplied by
.
Thus, mixed numbers may be represented by improper fractions, and improper fractions may be represented by mixed numbers.
Converting improper fractions to mixed numbers
To understand how we might convert an improper fraction to a mixed number, let's consider the fraction,
.
Thus,
.
We can illustrate a procedure for converting an improper fraction to a mixed number using this example. However, the conversion is
more easily accomplished by dividing the numerator by the denominator and using the result to write the mixed number.
Converting an improper fraction to a mixed number
To convert an improper fraction to a mixed number, divide the numerator by the denominator.
The whole number part of the mixed number is the quotient.
The fractional part of the mixed number is the remainder written over the divisor (the denominator of the improper fraction).
Sample set a
Convert each improper fraction to its corresponding mixed number.
Divide 5 by 3.
The improper fraction
.
. Divide 46 by 9.
The improper fraction
.
. Divide 83 by 11.
The improper fraction
.
Divide 104 by 4.
The improper fraction
.
Practice set a
Convert each improper fraction to its corresponding mixed number.
62
Converting mixed numbers to improper fractions
To understand how to convert a mixed number to an improper fraction, we'll recall
mixed number = (natural number) + (proper fraction)
and consider the following diagram.
Recall that multiplication describes repeated addition.
Notice that
can be obtained from
using multiplication in the following way.
Multiply:
Add:
. Place the 5 over the 3:
The procedure for converting a mixed number to an improper fraction is illustrated in this example.
Converting a mixed number to an improper fraction
To convert a mixed number to an improper fraction,
Multiply the denominator of the fractional part of the mixed number by the whole number part.
To this product, add the numerator of the fractional part.
Place this result over the denominator of the fractional part.
Sample set b
Convert each mixed number to an improper fraction.
Multiply:
.
Add:
.
Place 47 over 8:
.
Thus,
.
Multiply:
.
Add:
.
Place 50 over 3:
Thus,
Practice set b
Convert each mixed number to its corresponding improper fraction.
Exercises
For the following 15 problems, identify each expression as a proper fraction, an improper fraction, or a mixed number.
improper fraction
proper fraction
mixed number
improper fraction
mixed number
mixed number
proper fraction
mixed number
For the following 15 problems, convert each of the improper fractions to its corresponding mixed number.
or
For the following 15 problems, convert each of the mixed numbers to its corresponding improper fraction.
Why does
not qualify as a mixed number?
See the definition of a mixed number.
Why does 5 qualify as a mixed number?
See the definition of a mixed number.
… because it may be written as
, where
is any positive whole number.
Exercises for review
(
[link] ) Round 2,614,000 to the nearest thousand.
(
[link] ) Determine if 41,826 is divisible by 2 and 3.
(
[link] ) Find the least common multiple of 28 and 36.
252
(
[link] ) Specify the numerator and denominator of the fraction
.
Questions & Answers
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