Question

a fraction becomes 9/11 of 2 is added to denominator and numerator if 3 is added to both numerator and denominator it becomes 5/4 find fraction ​

Answer 1

Question:

A fraction becomes 9/11 if 2 is added to the denominator and numerator and if 3 is added to both numerator and denominator it becomes 5/6. Find the fraction.

Answer:

$\bigstar{\bold{The\:fraction=\dfrac{7}{9} }}$

Step-by-step explanation:

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

• A fraction becomes 9/11 if 2 is added to both numerator and denominator
• A fraction becomes 5/6 if 3 is added to both numerator and denominator.

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

• The fraction

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

→ Let the numerator of the fraction be x

→ Let the denominator of the fraction be y

→ Hence,

  The fraction = x/y

→ In the first case, the fraction becomes 9/11 if 2 is added to both numerator and    denominator

→ Hence,

  $\sf{\dfrac{x+2}{y+2} =\dfrac{9}{11} }$

→ Cross multiplying,

   11 ( x + 2) = 9(y + 2)

   11x + 22 = 9y + 18

   9y = 11x + 22 - 18

   9y = 11x + 4

   9y - 11x = 4 ------(1)

→ In the second case, if 3 is added to the numerator and denominator, the fraction becomes 5/6

→ Hence,

  $\sf{\dfrac{x+3}{y+3} =\dfrac{5}{6} }$

→ Cross multiplying,

   6 (x + 3) = 5 (y + 3)

   6x + 18 = 5y + 15

   6x - 5y = -3

   5y - 6x = 3---(2)

→ Multiply equation 1 by 5 and equation 2 by 9

  45y - 55x = 20

  45y - 54x = 27

→ Solving by elimination method,

          -x = -7

           x = 7

→ Hence the numerator of the fraction is 7

→ Substitute the value of x in equation 1

  9y - 11 × 7 = 4

  9y -77 = 4

  9y = 81

    y = 9

→ Hence the denominator of the fraction is 9

→ Therefore,

   The fraction = 7/9

$\boxed{\bold{The\:fraction=\dfrac{7}{9} }}$

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

$\sf{\dfrac{x+2}{y+2} =\dfrac{9}{11} }$

(7 + 2)/(9 + 2) = 9/11

9/11 = 9/11

$\sf{\dfrac{x+3}{y+3} =\dfrac{5}{6} }$

(7 + 3)/(9 + 3) = 5/6

10/12 = 5/6

5/6 = 5/6

→ Hence verified.

Answer 2

Information :- This solution is based on Cross Multiplication.

Given :-

Fraction becomes 9/11 if 2 is added to both numerator and denominator.

If 3 is added to both numerator and denominator it becomes 5/6.

Solution:-

Let the Numerator be x                  

Let the Denominator be y                  

Fraction = x/y                  

According to the Question,                  

⇒ (x + 2)/y + 2 = 9/11                  

On Cross multiplying,                  

⇒ 11x + 22 = 9y + 18                  

Subtracting 22 from both sides,                  

⇒ 11x = 9y – 4                  

Dividing by 11, we get                  

⇒ x = 9y – 4/11 … (i)                  

Then,              

⇒ (x+3)/y +3 = 5/6 … (ii)                  

On Cross multiplying,                  

⇒ 6x + 18 = 5y + 15                  

Subtracting the value of x, we get                  

⇒ 6(9y – 4 )/11 + 18 = 5y + 15                  

Subtracting 18 from both the sides                  

⇒ 6(9y – 4 )/11 = 5y – 3                  

⇒ 54 – 24 = 55y – 33                  

⇒ –y = – 9                  

⇒ y = 9                  

Putting this value of y in equation (i), we get                  

⇒ x = 9y – 4                  

⇒ x = (81 – 4)/77                  

⇒ x = 77/11                  

⇒ x = 7                  

Hence, the  fraction is 7/9.