Page 77 - ArithBook5thEd ~ BCC

3.3.3 Converting a mixed or whole number to an improper fraction

Sometimes we need to change a mixed number back into an improper fraction. The key fact, again, is
that the fraction
a
a
is always equal to 1, for any number a except 0. For example,

3
=1.
3

1
Example 82. Convert the mixed number 2 to an improper fraction, using the fact that 3 =1.
3 3
Solution. Think of the whole number 2 in the following way:

3 3
2= 1 + 1 = + .
3 3
1
It follows that the mixed number 2 can be written
3
3 3 1
+ + .
3 3 3
7
The figure below should make it clear that this sum is equal to the improper fraction .
3

+ + = 7
3

1= 3 1= 3 1
3 3 3

(The figure also demonstrates that fractions with the same denominator add up to a new fraction, with
the same denominator, and a numerator which is the sum of all the old numerators. We’ll say more
about adding fractions later. )

The procedure in the previous example is easily turned into a general formula.

p N · q + p
The mixed number N is equal to the improper fraction .
q q

Remember to follow the order of operations (multiplication before addition) when evaluating N · q + p.
2
Example 83. Convert the mixed number 8 into an improper fraction, using the boxed rule.
3

Page 77