Question

The denominator of a rational number is greater than its numerator by \[6\]. If the numerator is increased by \[5\] and the denominator is decreased by \[3\], the number obtained is \[\dfrac{5}{4}\]. Find the rational number.
A) \[\dfrac{2}{9}\]
B) \[\dfrac{5}{{11}}\]
C) \[\dfrac{7}{6}\]
D) \[\dfrac{{10}}{{11}}\]

Answer 1

Hint: We have to find the required rational number by using the given information about the rational number. First, we will consider the numerator. Then we find the denominator in terms of the numerator. Using the given information, we will find the rational number. Solving the equation, we will get the original rational number.

Complete step-by-step answer:
It is given that; the denominator of a rational number is greater than its numerator by \[6\]. If the numerator is increased by \[5\] and the denominator is decreased by \[3\], the number obtained is \[\dfrac{5}{4}\].
We have to find the rational number.
Let us consider, the numerator of the rational number is \[x\].
Since, the denominator of a rational number is greater than its numerator by \[6\]. The denominator is \[x + 6\].
So, the rational number is \[\dfrac{x}{{x + 6}}\].
Again, the numerator is increased by \[5\] and the denominator is decreased by \[3\].
The numerator will be \[x + 5\]and the denominator is \[x + 6 - 3 = x + 3\]
So, the rational number will be \[\dfrac{{x + 5}}{{x + 3}}\]
According to the problem,
$\Rightarrow$\[\dfrac{{x + 5}}{{x + 3}} = \dfrac{5}{4}\]
By cross multiplication we get,
$\Rightarrow$\[4(x + 5) = 5(x + 3)\]
Multiplying the terms,
$\Rightarrow$\[4x + 20 = 5x + 15\]
Rearranging the terms we get,
$\Rightarrow$\[4x - 5x = 15 - 20\]
Subtracting the terms,
$\Rightarrow$\[ - x = - 5\]
Simplifying again we get,
$\Rightarrow$\[x = 5\]
So, the numerator is \[x = 5\] and the denominator is \[5 + 6 = 11\]
Hence, the rational number is \[\dfrac{5}{{11}}\].

$\therefore $ The correct option is B) \[\dfrac{5}{{11}}\].

Note: We observe that in mathematics, a rational number is a number that can be expressed as the quotient or fraction $\dfrac{p}{q}$ of two integers, where p is the numerator and q is the non-zero denominator. Any fraction with non-zero denominators is a rational number. Any whole number is also a rational number.