BMCC: Math Tutorials: Fractions:
Improper Fractions and Mixed Numbers
Quick links to topics on this page:
Introduction to Fractions
Writing an Improper Fraction as a Mixed Number
Writing a Mixed Number as an Improper Fraction
Fractions are representations of parts of a whole. A "whole" can be almost anything-- something physical, like a pizza or a pound of cement, or something more abstract, like a circle or an amount of time. Whatever is being used as the "whole" for a particular example, consider that the whole is divided into some number n of portions, all of equal size. A fraction is a representation of some number of those portions compared to the original amount of the whole.
Suppose I have some exotic imported coffee. Using a kitchen scale, I divide the entire amount into 4 equal portions and put each portion into a separate plastic bag. I can represent the amount of coffee in each bag with a fraction: each bag contains one-fourth of the coffee. The second part of the fraction, the "fourth," indicates how into how many total portions the coffee was split. The "one" part of the fraction tells how many of those portions are in each bag. If I give 3 bags to you, then you have three-fourths of the coffee, and I still have one-fourth.
To express numerically the amount of coffee I gave you, we represent a fraction like this:
The line separating the upper number from the lower number is called the fraction bar. The number above the fraction bar is called the numerator, and the number below the bar is called the denominator. The denominator represents into how many equal portions the original "whole" was divided, while the numerator indicates how many of those portions are being considered. To indicate how much of my coffee you received, the fraction shows that the original amount was divided into 4 equal portions, and you now have 3 of those portions.
A proper fraction is one in which the numerator is smaller than the denominator, so a proper fraction has a value less than 1. is a proper fraction.
An improper fraction is one in which the numerator is greater than or equal to the denominator. An improper fraction is thus greater than or equal to 1. For instance, the three bags of coffee I gave you might have a total weight of 
pounds.
A mixed number is a combination of a whole number and a fractional part. The weight of the coffee in pounds that we just represented with an improper fraction can also be expressed as a mixed number:
In the above relation, 
is a mixed number.
As stated above, an improper fraction has a value equal to or greater than 1. If the numerator and denominator of a fraction are equal, then the fraction is equal to 1. If the numerator is greater than the denominator, then the fraction is greater than 1. Any fraction greater than 1 can be expressed as a mixed number.
Recall that a mixed number has two parts: a whole number part, and a fractional part. Converting an improper fraction to a mixed number has two steps, to find the two parts of the equivalent mixed number.
For instance, consider the improper fraction 
. This fraction is converted to a mixed number as follows:
For step 1, the numerator is divided by the denominator. In this case, we want to divide 11 by 7.
The quotient in this division problem is 1, which will be the whole number part of the mixed number. Now, for step 2, to determine the fractional part, we simply make a new fraction with a numerator equal to the remainder after division. The remainder in this problem is 4. The denominator for the new fraction will be the same as in the original improper fraction-- in this case, it is 7. The fractional part of the mixed number is thus 
. The mixed number is obtained by combining the whole number part and the fractional part, which gives 
.
Let's try another example. Express the improper fraction 
as a mixed number.
Step 1: We determine the whole number part of the mixed number by dividing the numerator of the improper fraction (17 in this example) by the denominator (5 in this example).
The whole number part of the mixed number is the quotient, which is 3 in this case. In step 2, we make the fractional part of the mixed number by putting the remainder from the above division (2 in this case) in the numerator, and keeping the denominator the same as in the original improper fraction (5 in this case), which yields 
as the fractional part. The mixed number is formed by combining the whole number part and the fractional part, to give 
.
When converting a mixed number to an improper fraction, the denominator of the improper fraction will be the same as the denominator of the fractional part of the mixed number. Determining the numerator of the mixed fraction has two steps.
It's easier than it sounds. Consider the mixed number 
. The improper fraction will have the same denominator as the fractional part of the mixed number, so the denominator will be 6. Now we determine the numerator of the improper fraction in two steps. For step 1 we multiply the denominator of the fractional part, which is 6, by the whole number part, which is 2. This gives a product of 12. For step 2 we add this product to the numerator of the fractional part, which is 5, so we have 12 + 5 = 17. The numerator of the improper fraction is 17, and the denominator is 6, so the improper fraction is 
.
Another example: express the mixed number 
as an improper fraction.
The denominator of the improper fraction will be the same as the denominator of the fractional part of the mixed number, which is 14. The numerator of the improper fraction is determined in two steps.
Step 1: Multiply the denominator of the fractional part of the mixed number (14) by the whole number part of the mixed number (3). This gives a product of 14 x 3 = 42.
Step 2: Add the product from step 1 to the numerator of the fractional part of the mixed number (11). This gives a sum of 42 + 11 = 53, so the numerator of the improper fraction will be 53. The improper fraction is thus 
.
Next: Equivalent Fractions
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