4.2+What+is+a+Fraction?

Section 4.2 - What is a Fraction?
A fraction represents a part of a whole. If we say we have one third of a pie, we take a whole pie, cut it into three parts, and we have one of those three parts, or one third of the whole pie.





Three thirds, then, means that we have the the pie cut into thirds and we have all three pieces, or one whole pie. 3/3 = 1.


 * Division by Zero Again**

The line between the numerator and denominator looks suspiciously like a division sign. Well, it is division. If you divide the numerator by the denominator, you will get the value of the fraction in a decimal form. Next chapter.

So, if we use the analogy of cutting the pie again, consider the two fractions below.

For the first fraction, the denominator suggests that the pie is being cut into three pieces and the numerator says we are getting none of those three pieces.

However the denominator on the second fraction says to cut the pie into zero pieces. That isn't possible. At a minimum, the pie will have 1 big piece (whole). So this is impossible (say "undefined"). If you cut a pie into zero pieces, how would you have 3 of them anyway?

Division by zero is "undefined".


 * Proper vs. Improper Fractions**


 * A Proper Fraction is one where the numerator is smaller than the denominator. **


 * An Improper Fraction is one where the numerator is either the same as the denominator, or larger than the denominator. **

Positive and negative signs have no effect on whether a fraction is considered "proper" or "improper".



Improper fractions are improper because they aren't a part of a whole - they are actually more than a whole. Doing math with improper fractions, however, is just the same as with proper fractions.


 * Note**: The Khan Academy video below defines proper fractions as equal to or less than 1. However, most texts and experts (including our department) say that a fraction equal to 1 is improper.

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 * Converting Whole Numbers to Fractions**

Any number, divided by 1, is itself. That means that you can turn any whole number into a fraction simply by making it the numerator and adding a denominator of 1.



Sometimes this will be valuable, particularly when multiplying or dividing fractions.


 * Mixed Numbers**

A mixed number is one that has a whole number part, and a fraction part. The fraction will always be a proper fraction, meaning less than one. Examples of mixed numbers:




 * Converting Mixed Numbers to Improper Fractions**

Mixed numbers are easy enough to comprehend ( 4 and a third (1/3) is a little more than 4 alone), but they are difficult to do math with, except for addition. Mixed numbers can be converted to improper fractions, which can then be used to multiply, divide, add, and subtract.

The converting is described like this: **Denominator X whole number + numerator, all over the denominator.**



If you have mixed numbers that are negative, ignore the negative while you do the conversion, but don't forget to put it back in front of the improper fraction.


 * Converting Improper Fractions to Mixed Numbers**

Divide the numerator by the denominator.

The answer will be the whole number, and the remainder will be the new numerator.



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 * Rounding a Mixed Number**

If you are rounding a mixed number, you are actually just rounding the fraction part to find the nearest whole number. The whole number part of the mixed number will either stay the same, or go up by 1. In order to do this, you need to decide if the fraction is less than half, or more than half. (If it is exactly half, round up.)

Three ways to decide if a fraction is more or less than one half: A. Double the numerator and see if it makes it an improper fraction. If it is, the fraction was more than a half and the whole number will be upped by 1. B. Is the numerator more than half of the denominator? If so, it is more than one half, so round up. C. You could compare it with one half and get common denominators (later section) for both fractions and see which is bigger. (D. Next chapter you could turn it into a decimal by dividing and see if it is more or less than 0.5)




 * Chapter Four Practice Problems**

**Where to from here?**
4.1 Greatest Common Factor 4.2 What is a Fraction? 4.3 Equivalent Fractions 4.4 Multiplying Fractions 4.5 Dividing Fractions 4.6 Adding and Subtracting Like Fractions 4.7 Adding and Subtracting Unlike Fractions 4.8 Fraction Coefficients 4.9 Solving for X with Fractions Chapter Four Review Chapter Four Homework