Question

What is \[\left( {\dfrac{{22}}{5}} \right)\] as a mixed fraction?

Answer 1

Hint: We first try to explain the mixed fraction and the representation in improper fraction. We use variables to express the condition between those representations. Then, we apply the long division method to express the given improper fraction as a mixed fraction.

Complete step by step answer:
The given fraction \[\left( {\dfrac{{22}}{5}} \right)\] is an improper fraction. Improper fractions are fractions in which the value of the numerator is greater than that of the denominator.
So, we have to convert the improper fraction \[\left( {\dfrac{{22}}{5}} \right)\] into a mixed fraction.
Mixed fractions are those fractions that have an integral value along with the fraction consisting of the numerator and denominator.
We express the process of conversion of improper fraction into mixed fraction in the form of variables.
Let the mixed fraction be $\left( {\dfrac{a}{b}} \right)$ where a and b are integers. Now, we express it in the form of a mixed fraction as $\left( {x\dfrac{y}{b}} \right)$ where b is greater than y and that follows $a = bx + y$.
Then, the condition for both mixed and improper fractions to be equal is: $\left( {x\dfrac{y}{b}} \right) = \left( {\dfrac{a}{b}} \right)$.
 The mixed fraction $\left( {x\dfrac{y}{b}} \right)$ can also be represented as $\left( {\dfrac{{xb + y}}{b}} \right)$.
Hence, $\left( {\dfrac{{xb + y}}{b}} \right) = \left( {\dfrac{a}{b}} \right)$.
We can also say that we obtain y as the remainder and x as the quotient when we divide a by b.
 So, the given question requires us to write \[\left( {\dfrac{{22}}{5}} \right)\] as a mixed fraction.
So, we will first divide the number \[22\] by $5$ and then substitute the values of quotient and remainder of the division into the representation of the mixed fraction.
So, we have,
\[5\overset{4}{\overline{\left){\begin{align}
  & 22 \\
 & \underline{20} \\
 & 2 \\
\end{align}}\right.}}\]
Hence, we get quotients as $4$ and remainder as $2$.
So, we can convert the improper fraction \[\left( {\dfrac{{22}}{5}} \right)\] into a mixed fraction as \[\left( {4\dfrac{2}{5}} \right)\].

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 is 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. The remainder will be the numerator of the mixed fraction.