Question

10.) The sum the numerator and the
denominator of a fraction is 4 more than
twice the numerator. If 3 is added to both
the numerator and the denominator, the
fraction reduces to 2/3. Find the fraction.
​

Answer 1

SoluTion :-

$\textsf {Let the numerator be x}\\\\\textsf {and denominator be y}$

$\therefore \ \boxed {\sf {Fraction = \dfrac{x}{y} }}$

According to the 1st condition,

$\textsf {The sum of the given x and y = Twice the numerator + 4}$

$\sf {x+y=2x+4}\\\\\\\sf {x+y-2x-4=0}\\\\\\\sf {-x+y=4}\\\\\\\sf {x-y=-1 \ .... (Equation \ 1) }$

According to the 2nd condition,

$\sf {x \ and \ y \ must\ be\ increased\ by\ 3\ to\ get \ the \ fraction \ \frac{2}{3} }$

$\sf { \frac { n + 3 } { d + 3 } = \frac { 2 } { 3 } }\\\\\\\sf { 3 n + 9 = 2 d + 6 }\\\\\\\sf { 3 n - 2 d = 6 - 9 }\\\\\\\sf { 3 n - 2 d = - 3 \ .... (Equation \ 2 ) }$

Substituting the value of y from Equation 1 to 2,

$\sf {3x-2(x+4)=-3}\\\\\\\sf {3x-2x+8=-3}\\\\\\\sf {x=5}$

Now,

$\sf {y=5+4}\\\\\\\sf {y=9}$

Thus,

$\sf {Answer - \boxed {\sf {Fraction = \dfrac{5}{9}}}}$

Answer 2

The given fraction will be 5/9.

Solution:

Let n be the numerator and d be the denominator.

Given,  

The sum of the given n and d is equal to twice the numerator plus 4, i.e.

→n + d = 2n + 4

→ -n + d = 4

→ n - d = -4.....(1)

From given, the numerator n and the denominator d must be increased by 3 to get the ratio 2:3

→n+3/d+3= 2/3

→3n-2d= 6-9

→3n-2d = -3

From equations (i) and (ii), we get,

(i) ×3 = 3n - 3d = -12

(ii) → 3n - 2d =-3

Solving (i) and (ii) we get,

d = -9

Substituting d=9 in equation (i), we get,

n-d = -4

n-9=-4

n= 5

∴ The given fraction is

n/d = 5/ 9