In a previous lesson, we reviewed fractions and equivalent fractions. In this lesson, we will discuss different kinds of fractions.
A proper fraction is classified as such when the numerator of a fraction has a lower value than the denominator.
Here are a few examples of proper fractions:
\(\cfrac{1}{4}, \cfrac{5}{9}, \cfrac{1}{2}, \cfrac{2}{3}, \cfrac{49}{586}\)
An improper fraction is classified as such when the numerator has a greater value than the denominator.
Here are a few examples of improper fractions:
\(\cfrac{5}{4}, \cfrac{7}{3}, \cfrac{9}{2}, \cfrac{17}{9}, \cfrac{365}{11}\)
In a previous lesson, we demonstrated how to reduce proper fractions and the same can be done with improper fractions.
Reduce the improper fraction \(\cfrac{250}{100}\).
First, as with proper fractions, we can divide both the numerator and denominator by a common denominator (in this case, \(10\):
\(\cfrac{250/10}{100/10} = \cfrac{25}{10}\)
Next, we can continue reducing both sides. In this instance, we will divide both the numerator and denominator by \(5\):
\(\cfrac{25/5}{10/5} = \cfrac{5}{2}\)
Now we cannot reduce this fraction any further. Since the numerator is \(5\) and the denominator is \(2\), their greatest common factor is \(1\). Any further dividing would simply result in more fractions. Therefore, we have reduced the fraction to its lowest possible terms.
Therefore, we can determine that \(\cfrac{250}{100}\) reduced to its lowest terms is \(\boldsymbol{\cfrac{5}{2}}\).
NOTE: When we discussed fractions in the previous lesson, we suggested trying to divide both the numerator and the denominator by just the numerator. With improper fractions, it is different since you divide both sides by the denominator instead.
Reduce the improper fraction \(\cfrac{60}{15}\).
Using the technique outlines above, we will divide both the sides by the denominator, \(15\):
\(\cfrac{60/15}{15/15} = \cfrac{4}{1}\)
As with any value where the denominator is \(1\), the value of the reduced fraction is the same as the numerator:
\(\cfrac{4}{1} = 4\)
Thereofe, we can determine that \(\cfrac{60}{15}\) reduced to its lowest terms is \(\boldsymbol{4}\).
In order to reduce this fraction, we will need to divide the numerator and denominator their greatest common factor:
\(36: 36, 1, 18, 2, 12, 3, 9, \boldsymbol{4}, 6\)
\(20: 20, 1, 10, 2, 5, \boldsymbol{4}\)
Since the greatest common factor is 4, we will divide both sides by \(4\):
\(\cfrac{36/4}{20/4} = \cfrac{9}{5}\)
Therefore, we can determine that \(\cfrac{36}{20}\) reduced to its lowest terms is \(\boldsymbol{\cfrac{9}{5}}\).
A mixed fraction is another way to write down a fraction where its quantity is greater than \(1\). It is denoted by a whole number followed by a proper fraction.
Here are a few examples of mixed fractions:
\(1\cfrac{1}{4}, 2\cfrac{1}{3}, 1\cfrac{8}{9}, 33\cfrac{2}{11}\)
The value of mixed fractions can be converted to an equivalent value of an improper fraction.
ExampleConvert the mixed fraction \(2\cfrac{5}{8}\) into an improper fraction.
First, we will multiply the denominator by the whole number:
\(8 \times 2 = 16\)
Next, we will add that value to the numerator from the mixed fraction:
\(16 + 5 = 21\)
The resulting value is \(21\) which represents the numerator for the improper fraction. We can now write the improper fraction with the original denominator (\(8\) and without a whole number. This gives us the following:
\(2\cfrac{5}{8} = \cfrac{21}{8}\)
Therefore, we can determine that \(2\cfrac{5}{8}\) converted to an improper fraction is \(\boldsymbol{\cfrac{21}{8}}\).
Convert the improper fraction \(\cfrac{31}{6}\) into a mixed fraction.
First, we can determine how many times \(6\) goes into \(31\). We know the following:
\(6 \times 5 = 30 \lt 31\)
We also know the following:
\(6 \times 6 = 36 \gt 31\)
As a result, we can determine that \(6\) goes into \(31\) a total of \(5\) times.
We now subtract the product of \(30\) from the numerator \(31\) to get the remainder:
\(31 - 30 = 1\)
We now use the value of \(5\) as the whole number, and the remainder of \(1\) as the numerator for the fraction with \(6\) as its denominator:
\(\cfrac{31}{6} = 5\cfrac{1}{6}\)
Therefore, we can determine that \(\cfrac{31}{6}\) converted to a mixed fraction is \(\boldsymbol{5\cfrac{1}{6}}\).
First we can multiply the whole number by the denominator:
\(6 \times 3 = 18\)
Then we add that value to the numerator (\(5\)):
\(18 + 5 = 23\)
The resulting value is \(23\) which represents the numerator for the improper fraction. We can now write the improper fraction with the original denominator (\(6\) and without a whole number. This gives us the following:
\(3\cfrac{5}{6} = \cfrac{23}{6}\)
Therefore, we can determine \(3\cfrac{5}{6}\) converted to an improper fraction is \(\boldsymbol{\cfrac{23}{6}}\)
First, we can determine how many times \(5\) goes into \(23\). We know the following:
\(5 \times 4 = 20 \lt 23\)
We also know the following:
\(5 \times 5 = 25 \gt 23\)
As a result, we can determine that \(5\) goes into \(23\) fully a total of \(4\) times.
We can now subtract \(20\) from the original numerator (\(23\)) to determine the remainder:
\(23 - 20 = 3\)
We can now use the value of \(4\) as the whole number and the remainder of \(3\) as the numerator:
\(4 \cfrac{3}{5}\)
Therefore, we can determine that \(\cfrac{23}{5}\) converted to a mixed fraction is \(\boldsymbol{4 \cfrac{3}{5}}\).