Question

The denominator of the rational number is greater than its numerator by 8.If numerator is increased by 17 and denominator is decreased by 1, the number obtained is $\dfrac{3}{2}$ . Find a rational number.

Answer 1

Hint: Start with a rational number then follow the question.

Complete step-by-step answer:

Let the required rational number is $\dfrac{p}{q}$ . The question says, the denominator of the rational number is greater than its numerator by 8. It means if we add 8 in the numerator then the numerator and the denominator will be equal. That is $p + 8 = q{\text{ - - - - - - - - - - (}}1{\text{)}}$
Now, if the numerator is increased by 17 and denominator is decreased by 1, the number obtained is $\dfrac{3}{2}$ . It means, $\dfrac{{p + 17}}{{q - 1}} = \dfrac{3}{2}{\text{ - - - - - - - - - - - - (2)}}$ . We have two equations and two unknowns. From equation one put the value of q in equation two.
\[\
  \dfrac{{p + 17}}{{q - 1}} = \dfrac{3}{2}{\text{ - - - - - - - - - - - - (2)}} \\
   \Rightarrow \dfrac{{p + 17}}{{p + 8 - 1}} = \dfrac{3}{2}{\text{ [}}q = p + 8{\text{]}} \\
   \Rightarrow {\text{2(}}p + 17{\text{) = 3(}}p + 7{\text{) [using cross multiplication]}} \\
   \Rightarrow 2p + 34 = 3p + 21 \\
   \Rightarrow 2p - 3p = 21 - 34 \\
   \Rightarrow { - }p ={ - }13 \\
   \Rightarrow p = 13 \\
\ \]
Now, we got the value of p and we use this value to get the value of q with the relation \[p + 8 = q\] as
$\
  p + 8 = q \\
   \Rightarrow 13 + 8 = q{\text{ }}[p = 13] \\
   \Rightarrow q = 21 \\
\ $
So, the required rational number is $\dfrac{p}{q} = \dfrac{{13}}{{21}}$ .

Note: In word problems like this we need to follow the question properly. We have to be careful when we are creating the equations. We need to form as many equations as unknowns. If it's not the case then something went wrong.