Question

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

Answer 1

Hint: Assuming the rational number to be $\dfrac{p}{q}$. Using the first condition that the denominator is greater than the numerator by 8 we get $q = p + 8$ and using the second condition we get $\dfrac{{p + 17}}{{q - 1}} = \dfrac{3}{2}$ cross multiplying and solving we get the value of p and q. Hence we get the required rational number.

Step by step solution :
We know that a rational number is of the form $\dfrac{p}{q}$
We are given that the denominator is greater than the numerator by 8
Here p is our numerator and q is our denominator
Hence we get ,$q = p + 8$ ……….(1)
Now we are given that when the numerator is increased by 17 and denominator is decreased by 1 we get $\dfrac{3}{2}$
When numerator is increased by 17 we get the numerator to be $p + 17$
When the denominator is decreased by 1 we get the numerator to be $q - 1$
Hence the fraction is $\dfrac{{p + 17}}{{q - 1}}$
Since it is equal to $\dfrac{3}{2}$ we get
$ \Rightarrow \dfrac{{p + 17}}{{q - 1}} = \dfrac{3}{2}$
Using (1) we get
$
   \Rightarrow \dfrac{{p + 17}}{{p + 8 - 1}} = \dfrac{3}{2} \\
   \Rightarrow \dfrac{{p + 17}}{{p + 7}} = \dfrac{3}{2} \\
   \Rightarrow 2\left( {p + 17} \right) = 3\left( {p + 7} \right) \\
   \Rightarrow 2p + 34 = 3p + 21 \\
   \Rightarrow 34 - 21 = 3p - 2p \\
   \Rightarrow 13 = p \\
 $
Hence we get the numerator to be 13
Using this in (1) we get
$
   \Rightarrow q = 13 + 8 \\
   \Rightarrow q = 21 \\
 $

Hence we get our rational number to be $\dfrac{{13}}{{21}}$.

Note:
1) Every fraction has a numerator that equals the number of parts we have and a denominator equaling the total number of parts in a whole.
2) A mixed number is the combination of a whole number and a fraction.
3) An improper fraction is a fraction whose numerator is larger than its denominator therefore having a value greater than one.