(i)
The denominator of a fraction is 1 less than twice its numerator. If 1 is added to numerator
and denominator respect

2 thoughts on “(i)<br />The denominator of a fraction is 1 less than twice its numerator. If 1 is added to numerator<br />and denominator respect”

1. Required Answer:–

   Let us consider that the numerator of the fraction is x and the denominator be y. Then the fraction will be [tex]\frac{x}{y}[/tex].

   According to question,

   The denominator of a fraction is 1 less than twice its numerator. That means,

   [tex]y = 2x – 1..(1)[/tex]

   We can keep this equation in this way. Later, we can try solving by substitution.

   Also,

   If 1 is added to numerator and denominator respectively, the ratio of numerator and denominator is 3 : 5. That means,

   [tex] \dfrac{x + 1}{y + 1} = \dfrac{3}{5} [/tex]

   Cross multiply to get an equation,

   [tex]5(x + 1) = 3(y + 1)[/tex]

   Opening the parentheses,

   [tex]5x + 5 = 3y + 3[/tex]

   [tex]5x – 3y + 2 = 0[/tex]

   Now from equation (1), substitute y = 2x – 1,

   [tex]5x – 3(2x – 1) + 2 = 0[/tex]

   Simplifying the above equation,

   [tex]5x – 6x + 3 + 2 = 0[/tex]

   [tex] – x + 5 = 0[/tex]

   [tex] – x = – 5[/tex]

   And,

   [tex]x = 5[/tex]

   Then,

   [tex]y = 2(5) – 1 = 9[/tex]

   And the required fraction will be [tex]\boxed{\frac{5}{9}}[/tex].

   How to check?

   • Now you can check both the situations provided in the question.

   • Adding 1 to both numerator and denominator gives 6/10 i.e. 3/5. Hence, verified too.

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2. Given :-

   Denominator of a fraction is 1 less than twice its numerator. If 1 is added to numerator and denominator respectively, the ratio of numerator and denominator is 3: 5.

   To Find :-

   The fraction

   Solution :-

   Let the numerator be x

   And denominator will be 2x – 1

   [tex]\sf \dfrac{x + 1}{2x – 1 + 1} = \dfrac{3}{5}[/tex]

   [tex]\sf\dfrac{x+1}{2x} = \dfrac{3}5[/tex]

   By cross multiplication

   3(2x) = 5(x + 1)

   6x = 5x + 5

   6x – 5x = 5

   x = 5

   Finding the fraction

   [tex]\sf Fraction =\dfrac{5}{2(5)-1}[/tex]

   [tex]\sf Fraction = \dfrac{5}{10-1}[/tex]

   [tex]\sf Fraction = \dfrac{5}{9}[/tex]

   Reply