Question

a fraction becomes ,if 2 is added to both numerator and denominaotor.if 3 is added is added to both numerator and denominator it becomes.find the fraction

Answer 1

Correct Question:

A 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. Find the fraction.

$\huge\mathfrak{Answer:}$

Given:

• We have been given that on adding 2 to both numerator and denominator of a fraction, it becomes 9/11.
• On adding 3 to both numerator and denominator of a fraction, it becomes 5/6.

To Find:

• We need to find the fraction.

Solution:

Let the numberator be x and denominator be y.

$\mapsto\sf{Fraction = \dfrac{x}{y}}$

Now, according to the question we have

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

On cross multiplying the terms, we have

$\sf{11x + 22 = 9x + 18}$

$\implies\sf{11x - 9y = 18 - 22}$

$\implies \sf{11x - 9y = - 4}$

$\implies\sf{11x = 9y - 4}$

$\implies\sf{x = \dfrac{9y - 4}{11}}$

Also, when 3 is added to both numerator and denominator, the fraction becomes 5/6. We have

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

$\implies\sf{6x + 18 = 5y + 15}$

$\implies\sf{6x - 5y = 15 - 18}$

$\implies\sf{6 \times \dfrac{9y - 4}{11 - 5y} = - 3}$

$\implies\sf{54y - 24 - 55y = - 3}$

$\implies\sf{ - y - 24 = - 3 \times 11}$

$\implies\sf{y + 24 = 33}$

$\implies\sf{y = 33 - 24}$

$\implies\sf{y = 9}$

When y = 9, therefore

$\sf{x = \dfrac{9 \times 9 - 4}{11}}$

$\implies\sf{x = \dfrac{77}{11}}$

$\implies\sf{x = 7}$

$\mapsto\sf{Fraction = \dfrac{7}{9}}$

Hence, the fraction is 7/9.

Answer 2

Correct Question:

A fraction becomes $\rm{\dfrac{9}{11}}$, if $\rm{2}$ is added to both the numerator and denominator. If $\rm{3}$ is added to both the numerator and denominator, it becomes $\rm{\dfrac{5}{6}}$. Find the fraction.

Answer:

❖ The fraction is $\rm{\dfrac{7}{9}}$.

Explanation:

→ Let the fraction be $\rm{\dfrac{x}{y}}$

※ It is given that, if 2 is added to both the numerator and denominator. So :

→ Numerator = (x + 2)

→ Denominator = (y + 2)

※ Required equation :

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

= $\rm{11(x + 2) = 9(y + 2)}$

= $\rm{11x + 22 = 9y + 18}$

= $\rm{11x - 9y = 18 - 22}$

= $\rm{11x - 9y = -4}$

= $\rm{x = 9y - \dfrac{4}{11}.....(1)}$

※ Adding 3 to both the numerator and denominator :

= $\rm{\dfrac{x + 3}{y + 3} = \dfrac{5}{6}.....(2)}$

= $\rm{6(x + 3) = 5(y + 3) }$

= $\rm{6x + 18 = 5y + 15}$

※ Putting and subtracting the value of x :

= $\rm{\dfrac{6(9y - 4)}{11} = 5y - 3}$

= $\rm{54 - 24 = 55y - 33}$

= $\rm{\cancel{-} y = \cancel{-} 9}$

= $\fbox{\rm y = 9}$

※ Putting the value of y in (2) :

= $\rm{x = 9y - 4}$

= $\rm{x = \dfrac{81 - 4}{11}}$

= $\rm{x = \cancel{\dfrac{77}{11}}}$

= $\fbox{\rm x = 7}$

※ Final answer :

• x = $\rm{7}$
• y = $\rm{9}$
• Fraction = $\rm{\dfrac{7}{9}}$

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬