A mixed number is a whole number together with a fraction; for instance, (that is, three and two-fifths) is a mixed number. This notation hides addition inside it; "three and two-fifths" actually means "three plus two-fifths".
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In particular, though the whole number and the proper fraction are placed next to each other, this placement does not represent multiplication!
We use mixed numbers because they are a more intuitive way of expressing the number of entire items, plus a fractional amount.
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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.
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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.
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.
Improper fractions can be converted into mixed numbers, and vice versa; mixed numbers can be converted into improper fractions.
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.
The standard way to convert a mixed number to an improper fraction is as follows:
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.
To do the conversion, I'll multiply the denominator (that is, the 16) by the whole number (that is, the 2) to get 32. Then I'll add the numerator (that is, the 3) to the 32 to get the new numerator (namely, 35). The denominator will remain the same; that is, the bottom number will remain 16.
Then my answer is:
To do the conversion, I'll multiply the denominator (that is, the 5) by the whole number (the 6) to get 30. Then I'll add the numerator (the 2) to the 30 to get the new numerator, 32. The denominator will remain the same; namely, 5.
So my answer is:
You can use the Mathway widget below to practice converting a percentage to a decimal. Try the entered exercise, or type in your own exercise. Then click the button to compare your answer to Mathway's. (Or skip the widget and continue with the lesson.)
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To go from an improper fraction to a mixed number, keep in mind that "fractions are division".
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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.
First, I do the long division to find the whole-number part (being the quotient) and the remainder:
The quotient, across the top, is 11, so this will be the whole-number portion of the mixed number. Since the remainder is 1 and I'm dividing by 4, the fractional part will be ¼:
You can use the Mathway widget below to practice converting an improper fraction to a mixed number. Try the entered exercise, or type in your own exercise. Then click the button to compare your answer to Mathway's. (Or skip the widget and continue with the lesson.)
Please accept "preferences" cookies in order to enable this widget.
(Clicking on "Tap to view steps" on the widget's answer screen will take you to the Mathway site for a paid upgrade.)
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):
First, I'll do the long division to find the regular polynomial part and the remainder:
The polynomial on top, being the x + 1, is the equivalent to the whole-number part of a mixed number. The remainder at the bottom, being the −1, will create the proper-fraction part of this mixed (polynomial) number. Since the original rational expression had a denominator of x + 2, then the fractional part of my polynomial mixed number will be .
Then my answer is:
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.
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