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Mixed Numbers and Improper Fractions

On many jobs if you work overtime, the rate of pay increases to one-and-a-half times the regular rate. A number such as with a whole number part and a proper fraction part, is called a mixed number. A mixed number can also be expressed as an improper fraction, that is, a fraction whose numerator is larger than or equal to its denominator. The number is an example of an improper fraction.

Explain why a fraction whose numerator is larger than or equal to its denominator must have a value greater than or equal to 1.

Diagrams help us understand that mixed numbers and improper fractions are different forms of the same numbers, as Example 1 illustrates.

EXAMPLE 1

Draw diagrams to show that .

Solution

First, represent the mixed number and the improper fraction in diagrams.

Both diagrams represent so the numbers and must be equal.

In Example 1 each unit (or square) corresponds to one whole, which is also three-thirds. That is why the total number of thirds in is (2 3) + 1, or 7. The number of wholes in is 2 wholes, with of a whole left over. We can generalize these observations into two rules.

To Change a Mixed Number to an Improper Fraction

  • multiply the denominator of the fraction by the whole number part of the mixed number,
  • add the numerator of the fraction to this product, and
  • write this sum over the original denominator to form the improper fraction.

EXAMPLE 2

Write each of the following mixed numbers as an improper fraction.

Solution

a. We first multiply the denominator 9 by the whole number 3, getting 27. We next add the numerator 2. We then write the sum 29 over the original denominator to get .

b.

To Change an Improper Fraction to a Mixed Number

  • divide the numerator by the denominator, and
  • if there is a remainder, write it over the denominator.

EXAMPLE 2

Write each improper fraction as a mixed or whole number.

Solution

Divide the numerator by the denominator.

Write the remainder over the denominator.

In other words, 5 R1 means that in there are 5 wholes with of a whole left over.

Changing an improper fraction to a mixed number is important when we are dividing whole numbers: It allows us to express any remainder as a fraction. Previously, we would have said that the problems both have the answer 3 R1. But by interpreting these problems as improper fractions, we see that their answers are different.

When a number is expressed as a mixed number, we know its size more readily than when it is expressed as an improper fraction. For instance, consider the mixed number . We immediately see that it is larger than 11 and smaller than 12 (that is, between 11 and 12). We could not reach this conclusion so easily if we were to examine only its improper form. However, there are situations—when we multiply or divide fractions—in which the use of improper fractions is preferable.

 
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