Question

the sum of the numerator and denominator of a fraction is 4 more thsn twice the numerator. if the numerator and denominator are increased by 3 , they are in the. ratio 2:3. Determine the fraction​

Answer 1

Answer:

$\bigstar{\bold{The\:fraction\:is\:\dfrac{5}{9} }}$

Step-by-step explanation:

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

• Sum of numerator and denominator = 4 + 2(numerator)
• If the numerator and denominator increased by 3, they are in the ratio    2 : 3

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

• The fraction

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

→ Let the numerator be x

→ Let the denominator be y

→ By given,

 x + y = 4 + 2x

 2x - x - y = -4

 x - y = -4

 x = -4 + y -----(1)

→ Also by given increasing numerator and denominator by 3, they are in the ratio 2 : 3

→ Hence,

  $\sf{\dfrac{x+3}{y+3}=\dfrac{2}{3} }$

→ Substitute equation 1 in the above equation,

  $\sf{\dfrac{-4+y+3}{y+3} =\dfrac{2}{3}}$

   $\sf{\dfrac{y-1}{y+3}=\dfrac{2}{3} }$

→ Cross multiplying,

  3 (y - 1) = 2(y + 3)

  3y - 3 = 2y + 6

  3y - 2y = 6 + 3

            y = 9

→ Hence the denominator of the fraction is 9

→ Substitute the value of equation 1

  x = -4 + y

  x = -4 + 9

  x = 5

→Hence the numerator of the fraction is 5

$\boxed{\bold{The\:fraction\:is\:\dfrac{5}{9} }}$

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

→ Sum of numerator and denominator = 4 + 2 (numerator)

  5 + 9 = 4 + 2 × 5

  14 = 4 + 10

  14 = 14

→ If the numerator and denominator increased by 3, they are in the ratio 2 :    3

  $\sf{\dfrac{x+3}{y+3}=\dfrac{2}{3}}$

  $\sf{\dfrac{5+3}{9+3} =\dfrac{2}{3} }$

  $\sf{\dfrac{8}{12}=\dfrac{2}{3} }$

→ Hence verified.