Question

If the numerator of a fraction is increased by \[200\% \] and the denominator is increased by \[350\% \], the resultant fraction is \[\dfrac{5}{{12}}\], what was the original fraction ?
A.\[\dfrac{5}{9}\]
B.\[\dfrac{5}{8}\]
C.\[\dfrac{7}{{12}}\]
D.\[\dfrac{{11}}{{12}}\]

Answer 1

Hint: Here we will use the concept of the fraction. First, we will assume the numerator and denominator to be some variable. Then find the value of the new numerator and the value of the new denominator by adding increased percentage. Then we will equate the new fraction form to the given value \[\dfrac{5}{{12}}\] and solve the equation to get the value of the original fraction.

Complete step-by-step answer:
Let the numerator of the original fraction be \[x\] and let the denominator of the original fraction be \[y\].
It is given that the numerator of a fraction is increased by \[200\% \]. Therefore, we will find the value of the new numerator formed. Therefore, we get
\[x + \left( {\dfrac{{200}}{{100}}} \right)x = 3x\]
It is also given that the denominator is increased by \[350\% \]. Therefore, we will find the value of the new denominator formed. Therefore, we get
\[y + \left( {\dfrac{{350}}{{100}}} \right)y = \dfrac{{45}}{{10}}y\]
Now according to the question the resultant fraction is equal to \[\dfrac{5}{{12}}\]. Therefore, by equating the new fraction to this value we will get the value of the original fraction. Therefore, we get
\[\dfrac{{3x}}{{\dfrac{{45}}{{10}}y}} = \dfrac{5}{{12}}\]
\[ \Rightarrow \dfrac{{30x}}{{45y}} = \dfrac{5}{{12}}\]
Now we will solve this above equation to get the value of the original fraction, we get
\[ \Rightarrow \dfrac{x}{y} = \dfrac{{5 \times 45}}{{12 \times 30}}\]
\[ \Rightarrow \dfrac{x}{y} = \dfrac{5}{8}\]
Hence the value of the original fraction is equal to \[\dfrac{5}{8}\].
So, option B is the correct option.

Note: We should know the basic three types of fractions. We know the definition of three main types of fraction i.e. proper fractions, improper fractions, and mixed fractions.
Proper fractions are a fraction having the numerator less, or lower in degree, than the denominator. The value of proper fraction after simplification is always less than 1.
An improper fraction is a fraction where the numerator is greater than or equals to the denominator. After the simplification of an improper fraction results in the value which is equal or greater than 1, but not less than 1.
A mixed Fraction is the combination of a natural number and fraction. After the simplification of a mixed fraction results in the value which is always greater than 1.