Mixed Numbers and Improper Fractions

In particular, though the whole number and the proper fraction are placed next to each other, this placement does not represent multiplication!

Why do we use mixed numbers?

We use mixed numbers because they are a more intuitive way of expressing the number of entire items, plus a fractional amount.

For example, suppose you had a big pizza party and, after everybody goes home, you find that you have one pineapple pizza and half an anchovy pizza left over. You would say that you have one and a half pizzas left over. "One and a half" is the standard spoken-English way of expressing this number, and it is written as "1½". This expression, "1½", is called a mixed number, because it combines the "regular" number 1 with the fraction ½.

While mixed numbers are the natural choice for spoken English (and are therefore well-suited to the answers of word problems), they aren't generally the easiest fractions to compute with. In algebra, you will almost always prefer that your fractions not be mixed numbers.

What is an improper fraction?

An improper fraction is one whose numerator is larger than its denominator; that is, an improper fraction has a larger number on top than underneath. In algebra (and other math and science classes), you will generally use improper fractions, rather than mixed numbers. Using improper-fraction form makes multiplication so much easier, because there is no hidden addition as there is in a mixed number.

Can improper fractions be converted into mixed numbers?

Improper fractions can be converted into mixed numbers, and vice versa; mixed numbers can be converted into improper fractions.

Why do we convert improper fractions to mixed numbers?

Mixed numbers are the intuitive way of expressing improper fractions, so we will convert an improper fraction to mixed-number form when we are done with our computation (for, say, a word problem) and want to state our answer into a form that is more easily comprehended.

For instance, if somebody asks you how long a certain lecture went, you are far more likely to say that it dragged on for "two and a half hours" rather than "it went five-halves of an hour". The two expressions, 2½ and , mean the exact same amount, but people more easily understand the expression "(some number of entire hours) plus (some additional portion of an hour)".

While we generally use improper fractions in our computations, we often convert our final answers to mixed numbers, especially when we're working in a "real world" context such as a word problem. In other words, the improper fractions are more math-friendly; the mixed numbers are more user-friendly. The mixed numbers reflect how our brains seem to be hard-wired to understand.

How do you convert a mixed numbers to an improper fraction?

The standard way to convert a mixed number to an improper fraction is as follows:

1. Take the number in the denominator; for example, in , you would take the 4 from the denominator.
2. Multiply the denominator by the whole number. Continuing this example, you would multiply the denominator's 4 by the whole number 2 to get 8.
3. To this product, add the number from the numerator. In this example, you would add the 3 to the 8 obtained in the previous step, to get 11.
4. Create the corresponding improper fraction by putting the result over the original denominator. In this example, you would put the 11 over the 4, to get the corresponding improper fraction .

For instance, to convert 4½ to an improper fraction, you would do the following:

I multiplied the bottom 2 by the whole number 4, and then added in the 1 from on top, getting 9. Then I put this 9 on top of the 2 from the bottom of the original fractional portion of the mixed number.

After a little practice, you'll probably be doing these steps in your head. The process really is that simple and straightforward.

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• Convert to an improper fraction.

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• Convert to an improper fraction.

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How do you convert an improper fraction to a mixed number?

To go from an improper fraction to a mixed number, keep in mind that "fractions are division".

1. Take the numerator and the denominator, and do the long division of the numerator by the denominator. For example, given the improper fraction , divide the 13 by the 4.
2. Stop dividing once you've reached the point where, if you went further, you'd be into decimal places. In our example, you'd stop once you had the 3 on top of the long division. This whole-number quotient will be the whole-number part of the mixed number.
3. Note the remainder; put this remainder over the original denominator as the fractional part of the mixed number. In our example, the remainder will be 1 and the original denominator was 4, so the mixed number will be 3¼.

Note: When you're converting from improper fraction to mixed numbers, do not continue the long division into decimal places. Just find the whole-number quotient and the remainder. Then stop.

• Convert to a mixed number.

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This procedure works perfectly well on rational expressions (that is, on polynomial fractions), too. You can see this in the example below (or else skip this topic and jump on ahead to multiplying regular fractions):

• Convert the (improper) rational expression to the sum of a polynomial and a proper rational expression.

Whether or not you want to convert the addition of a negative to the subtraction of a positive — that's entirely up to you, unless your instructor specifies otherwise. Either form means the same thing.