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: five divided by two, eight divided by five, 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 open two multiplication three close plus one equals seven, so therefore equals seven divided by three.

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: equation sequence open two multiplication three close plus one divided by three equals six plus one divided by three equals seven divided by three.

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

mixed number equation sequence equals left parenthesis whole multiplication denominator right parenthesis plus numerator divided by denominator equals improper fraction

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You can also change improper fractions back into mixed numbers. For example, for 17 divided by eight, 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, 17 divided by eight equals. You can also carry out the division:

Sum showing 17 over 8 = 17 divided by 8 = 8 divided by 17 – 16 = 1

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

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6.1.10 Practice with Fractions