Question

The numerator of a fraction is 4 less than the denominator. if 1 added to both it's numerator and denominator, it becomes 1/2. find the fraction​

Answer 1

Answer:

$\bigstar{\bold{Fraction=\dfrac{3}{7} }}$

Step-by-step explanation:

$\Large{\underline{\rm{Given:}}}$

• Numerator of the fraction is 4 less than the denominator
• If 1 is added to both the numerator and denominator, fraction becomes 1/2.

$\Large{\underline{\rm{To\:Find:}}}$

• The fraction

$\Large{\underline{\rm{Solution:}}}$

↬ Let the denominator of the fraction be x.

↬ Hence,

   Numerator = x - 4

↬ Therefore,

    $\tt{The\:fraction=\dfrac{x-4}{x}}$

↬ By given,

    Adding 1 to both numerator and denominator, it becomes 1/2.

   $\tt{\dfrac{x-4+1}{x+1} =\dfrac{1}{2} }$

     $\tt{\dfrac{x-3}{x+1}=\dfrac{1}{2}}$

↬ Cross multiplying it we get,

   2(x - 3) = x + 1

   2x - 6 = x + 1

   2x - x = 1 + 6

    x = 7

↬ Hence the denominator of the fraction is 7.

↬ Now,

   Numerator = x - 4

   Numerator = 7 - 4 = 3

↬ Hence numerator of the fraction is 3.

↬ Therefore the fraction is 3/7.

    $\boxed{\bold{Fraction=\dfrac{3}{7} }}$

$\Large{\underline{\rm{Verification:}}}$

↬ Numerator = Denominator - 4

   3 = 7 - 4

   3 = 3

↬ Adding 1 to both numerator and denominator, the fraction becomes 1/2.

   (3 + 1)/(7+1) = 1/2

   4/8 = 1/2

   1/2 = 1/2

Hence verified.

Answer 2

$\sf{\bold{\green{\underline{\underline{Given}}}}}$

• Numberator is 4 less than denominator
• If one is added to both the fractions becomes 1 / 2

______________________

$\sf{\bold{\green{\underline{\underline{To\:Find}}}}}$

• Find the fraction = ??

______________________

$\sf{\bold{\green{\underline{\underline{Solution}}}}}$

⠀⠀⠀⠀

let the denominator be x

⠀⠀⠀⠀

ATQ : -

⠀⠀⠀⠀

numerator = x - 4

⠀⠀⠀⠀

• If one is added to both the fractions becomes 1 / 2

⠀⠀⠀⠀

$\sf:\implies \: {\bold{ \dfrac{x-4+1}{x+1} = \dfrac{1}{2} }}$

⠀⠀⠀⠀

$\sf:\implies \: {\bold{ \dfrac{x- 3}{x+1} = \dfrac{1}{2} }}$

⠀⠀⠀⠀

$\sf:\implies \: {\bold{ 2 ( x - 3 ) = 1 ( x + 1 )}}$

⠀⠀⠀⠀

$\sf:\implies \: {\bold{ 2x - 6 = x + 1 }}$

⠀⠀⠀⠀

$\sf:\implies \: {\bold{ 2x - x = 6+ 1}}$

⠀⠀⠀⠀

$\sf:\implies \: {\bold{ x = 7 }}$

⠀⠀⠀⠀

$\sf{\red{\boxed{\bold{Denominator = 7}}}}$

⠀⠀⠀⠀

• Finding numerator

⠀

numerator = x - 4

⠀⠀⠀⠀

numerator = 7 - 4

⠀⠀⠀⠀

numerator = 3

⠀⠀⠀

______________________

$\sf{\bold{\green{\underline{\underline{Answer}}}}}$

• Required fraction = $\sf{\bold{ \dfrac{3}{7} }}$
• ⠀⠀⠀⠀

⠀⠀⠀⠀

Verification :-

⠀⠀⠀⠀

$\sf:\implies \: {\bold{ \dfrac{3+1}{7+1} = \dfrac{1}{2} }}$

⠀⠀⠀⠀

$\sf:\implies \: {\bold{ \dfrac{4}{8} = \dfrac{1}{2} }}$

⠀⠀⠀⠀

$\sf:\implies \: {\bold{ \dfrac{1}{2} = \dfrac{1}{2} }}$

⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀.... Hence Proved