Question

the denominator of a rational number is greater than its numerator by 6. If the numerator is decreased by 1 , the denominator is increased by 1 , the number obtained is 1/3 . Find the rational number. ​

Answer 1

Working out:

In the above question, we have to find the fraction by considering the numerator and denominator any variable. And then, framing equations to get the required fraction.

Let,

• The numerator be x
• And the denominator be y

GiveN ATQ,

• Denominator = Numerator + 6

Can be written as,

⇛ y = x + 6 --------(1)

And,

• Numerator - 1/ Denominator + 1 = 1/3

Can be written as,

⇛ x - 1/y + 1 = 1/3

Cross multiplying,

⇛ 3(x - 1) = y + 1

⇛ 3x - 3 = y + 1

⇛ 3x - y = 4

Substituting y from equation (1) in equation (2),

⇛ 3x - (x + 6) = 4

⇛ 3x - x - 6 = 4

⇛ 2x - 6 = 4

⇛ 2x = 10

⇛ x = 10/2 = 5

Then, y = x + 6 = 11

Hence,

• Numerator of the fraction = 5
• Denominator of the fraction = 11

So, our required original fraction is:

$\huge{ \boxed{ \red{ \sf{ \frac{5}{11} }}}}$

And we are done !!

Answer 2

Answer:

Let the Numerator be a and Denominator be (a + 6) of the Fraction.

If 1 is subtracted from the numerator and 1 is added to the denominator, the number becomes 1/3

$\underline{\bigstar\:\textsf{According to the given Question :}}$

$:\implies\sf \dfrac{Numerator-1}{Denominator+1}=\dfrac{1}{3}\\\\\\:\implies\sf \dfrac{a-1}{(a+6)+1}=\dfrac{1}{3}\\\\\\:\implies\sf \dfrac{a-1}{a+7}=\dfrac{1}{3}\\\\\\:\implies\sf 3(a - 1) = 1(a + 7)\\\\\\:\implies\sf 3a - 3 = a + 7\\\\\\:\implies\sf 3a -a = 7 + 3\\\\\\:\implies\sf 2a= 10\\\\\\:\implies\sf a = \dfrac{10}{2}\\\\\\:\implies\sf a = 5$

⠀

$\dag\:\underline{\boxed{\sf Original\:Fraction=\dfrac{a}{a + 6} = \dfrac{5}{(5 + 6)} = \dfrac{5}{11} }}$