Question

How do you convert each improper fraction to a mixed fraction: $\left( {\dfrac{{67}}{{12}}} \right)$?

Answer 1

Hint: We first try to explain the improper fraction and the representation in mixed fraction. We use variables to express the condition between those representations. Then, we apply the long division method to represent the improper fraction into a mixed fraction consisting of a whole number and fractional part.

Complete step by step answer:
The given fraction $\left( {\dfrac{{67}}{{12}}} \right)$ is an improper fraction. Improper fractions are those fractions whose numerator is more in value than the denominator.
Mixed fractions are those fractions that have an integral value along with the fraction consisting of the numerator and denominator.
We need to convert the given improper fraction $\left( {\dfrac{{67}}{{12}}} \right)$ in which the value of numerator is greater than the denominator into mixed fraction which is representation in the form sum of a whole number and a proper fraction having numerator less than the denominator.
We express the process of conversion of improper fraction into mixed fraction in the form of variables.
Let the improper fraction be $\left( {\dfrac{p}{q}} \right)$ where p is greater than q. Now, we express it in the form of mixed fraction as $\left( {x\dfrac{a}{b}} \right)$ where x is an integer.
Then, the condition for both mixed and improper fractions to be equal is: $\left( {x\dfrac{a}{b}} \right) = \left( {\dfrac{p}{q}} \right)$.
 The mixed fraction $\left( {x\dfrac{a}{b}} \right)$ can also be represented as $\left( {\dfrac{{xb + a}}{b}} \right)$.
Hence, $\left( {\dfrac{{xb + a}}{b}} \right) = \left( {\dfrac{p}{q}} \right)$.
 So, the given question requires us to write $\left( {\dfrac{{67}}{{12}}} \right)$ as a mixed fraction.
So, we will make use of the long division process. The quotient obtained will be equal to the whole number of part of the mixed fraction and the remainder will be placed above the denominator representing the proper fractional part of the mixed fraction.
So, we have,
\[12\overset{5}{\overline{\left){\begin{align}
  & 67 \\
 & \underline60 \\
 & 7\\
\end{align}}\right.}}\]
So, we get the quotient as $5$ and remainder as $7$.
Therefore, we can represent the improper fraction $\left( {\dfrac{{67}}{{12}}} \right)$ as $\left( {5\dfrac{7}{{12}}} \right)$
Hence, the mixed fraction representation is $\left( {5\dfrac{7}{{12}}} \right)$.

Note:
We need to remember that the denominator in both cases of improper fraction and the mixed fraction will be the same. The only change happens in the numerator. The above method involves the long division process where the denominator is the divisor, the numerator is the dividend and the integer of the mixed fraction is the quotient of the long division process. The remainder will be the numerator of the mixed fraction.