Question

The denominator of a rational number is greater than its numerator by 5. If the numerator is increased by 6 and denominator is decreased by 3, the number becomes \[\dfrac{5}{3}\]. Find the original number. Choose the correct option:
A. \[\dfrac{9}{4}\]
B. \[\dfrac{4}{9}\]
C. \[\dfrac{-4}{9}\]
D. None of the above.

Answer 1

Hint: Start with the rational number\[\dfrac{a}{b}\], then form the equations as per the statements given to get the value of a and b.

Complete step-by-step solution -
In the question, it is given that the denominator of a rational number is greater than its numerator by 5. So let the original rational number is \[\dfrac{a}{b}\]. So according to the question, we have:
\[\Rightarrow b=a+5………………….(eq\,\,1)\].
Now it is also given that if the numerator is increased by 6 and denominator is decreased by 3, the number becomes\[\dfrac{5}{3}\]. So the equation that we will get is:
\[\Rightarrow \dfrac{a+6}{b-3}=\dfrac{5}{3} ………….(eq\,\,2)\]
Now, using equation (1), we get;
\[\begin{align}
  & \Rightarrow \dfrac{a+6}{(a+5)-3}=\dfrac{5}{3} \\
 & \Rightarrow \dfrac{a+6}{a+2}=\dfrac{5}{3} \\
 & \Rightarrow 3(a+6)=5(a+2) \\
 & \Rightarrow (3a+18)=(5a+10) \\
 & \Rightarrow (8)=(2a) \\
 & \Rightarrow a=4 \\
\end{align}\]
Next from equation (1), we get b as follows:
\[\begin{align}
  & \Rightarrow b=a+5\,\, \\
 & \Rightarrow b=4+5\, \\
 & \Rightarrow b=9\, \\
\end{align}\]
So, the original rational number is \[\dfrac{a}{b}=\dfrac{4}{9}\].

Note: The rational number is of the form\[\dfrac{a}{b}\], here\[b\ne 0\]. So we can say that a fraction number is also a rational number. Also, any integer can be written in the rational number form.