6.1.9 Improper Fractions

A fraction in which the numerator is greater than or equal to the denominator is known as an improper fraction. You can think of these as “top-heavy” fractions: , , and the like.

Mixed numbers, such as can be rewritten as improper fractions. You can imagine this in terms of pizzas. If you have two and one third pizzas, and cut each of the whole pizzas into thirds, then how many thirds of a pizza will you have?

In this case, the total number of thirds will be , so therefore .

You don’t want to have to draw out a picture each time you need to convert a mixed number into an improper fraction. Think about how we found the seven thirds. Because there were two whole parts broken into thirds, plus 1 extra third, we multiplied 2 by 3, then added 1. Since these are all thirds, we placed the 7 over 3 in fractional notation. We can work it out like this: .

This works for any mixed number that we need to convert into an improper fraction. So, the rule is:

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You can also change improper fractions back into mixed numbers. For example, for , imagine you have some pizzas cut into eight equal slices (eighths), and that you have 17 slices. You know that eight slices make one whole pizza, and that two pizzas would be 16 slices. There would be one-eighth (one slice) left over. So, . You can also carry out the division:

Since eight goes into 17 at most two whole times, and there is one out of eight parts left over. This tell us .

Need a little refresher on long division? (Click on “View document”).

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6.1.10 Practice with Fractions