Question

How do you write \[\left( {10\dfrac{3}{5}} \right)\] as an improper 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 reverse method of the long division method to express the mixed fraction as an improper fraction.

Complete step by step solution:
The given fraction \[\left( {10\dfrac{3}{5}} \right)\] is a mixed fraction. 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 mixed fraction which is represented in the form sum of an integer and a proper fraction into an improper fraction in which the value of the numerator is greater than the denominator.
We express the process of conversion of mixed fraction into improper fraction in the form of variables.
Let the mixed fraction be $\left( {x\dfrac{a}{b}} \right)$ where x is an integer. Now, we express it in the form of improper fraction as $\left( {\dfrac{p}{q}} \right)$ where p is greater than q.
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 \[\left( {10\dfrac{3}{5}} \right)\] as an improper fraction.
So, this can be done by the following steps above mentioned.
Hence, \[\left( {10\dfrac{3}{5}} \right) = \left( {\dfrac{{5 \times 10 + 3}}{5}} \right)\]
Simplifying further and doing the calculations, we get,
\[ = \left( {\dfrac{{50 + 3}}{5}} \right)\]
Adding the numbers in numerator,
\[ = \left( {\dfrac{{53}}{5}} \right)\]
So, the mixed fraction \[\left( {10\dfrac{3}{5}} \right)\] can be represented as an improper fraction as \[\left( {\dfrac{{53}}{5}} \right)\].

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 reverse of 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.