ANSWER:

When we ADD 6 to the numerator of a fraction, we get 1/2.

When we add 7 to the denominator of the same fraction, we get 1/3.

⇝ To FIND :-

The Fraction by Forming 2 equations.

⇝ Solution :-

Let For the Original Fraction :

Numerator = x

Denominator = y

So Original Fraction is : \dfrac{\text x}{\text y}

y

x

❒ When 6 is added to The Numerator :

Fraction Becomes : \dfrac{\text x+6}{\text y}

y

x+6

★ According To Question :

\begin{gathered} \dfrac{\text x + 6}{\text y} = \frac{1}{2} \\ \end{gathered}

y

x+6

=

2

1

\begin{gathered}:\longmapsto2(\text x + 6) = \text y \\ \end{gathered}

:⟼2(x+6)=y

\begin{gathered}:\longmapsto2\text x + 12 = \text y \\ \end{gathered}

:⟼2x+12=y

\begin{gathered}:\longmapsto2\bf x - y = - 12 \: ----(1) \\ \end{gathered}

:⟼2x−y=−12−−−−(1)

❒ When 7 is added to The Denominator :

Fraction Becomes : \dfrac{\text x}{\text y+6}

y+6

x

★ According To Question :

\begin{gathered} \dfrac{\text x}{\text y + 7} = \frac{1}{3} \\ \end{gathered}

y+7

x

=

3

1

\begin{gathered}:\longmapsto3\text x = \text y + 7 \\ \end{gathered}

:⟼3x=y+7

\begin{gathered}:\longmapsto \bf 3x - y = 7 \: - - - - (2) \\ \end{gathered}

:⟼3x−y=7−−−−(2)

✏ Subtracting (1) From (2) :

\begin{gathered}\purple{ \Large :\longmapsto \underline {\BOXED{{\bf x = 19} }}} \\ \end{gathered}

:⟼

x=19

✏ Putting Value of x in (2) :

\begin{gathered}:\longmapsto3 \times 19 - \text y = 7 \\ \end{gathered}

:⟼3×19−y=7

\begin{gathered}:\longmapsto57 - \text y = 7 \\ \end{gathered}

:⟼57−y=7

\begin{gathered}:\longmapsto - \text y = 7 - 57 \\ \end{gathered}

:⟼−y=7−57

\begin{gathered}:\longmapsto \cancel - \text y = \cancel- 50 \\ \end{gathered}

:⟼

−

y=

−

50

\purple{ \Large :\longmapsto \underline {\boxed{{\bf y = 50} }}}:⟼

y=50

As,

Original Fraction = \dfrac{\text x}{\text y}

y

x

Hence,

\large\underline{\pink{\underline{\frak{\pmb{\text Original \:\: Fraction = \dfrac{19}{50} }}}}}

OriginalFraction=

50

19

OriginalFraction=

50

19

Explanation:

When we add 6 to the numerator of a fraction, we get 1/2.

When we add 7 to the denominator of the same fraction, we get 1/3.

⇝ To Find :-

The Fraction by Forming 2 equations.

⇝ Solution :-

Let For the Original Fraction :

Numerator = x

Denominator = y

So Original Fraction is : \dfrac{\text x}{\text y}

y

x

❒ When 6 is added to The Numerator :

Fraction Becomes : \dfrac{\text x+6}{\text y}

y

x+6

★ According To Question :

\begin{gathered} \dfrac{\text x + 6}{\text y} = \frac{1}{2} \\ \end{gathered}

y

x+6

=

2

1

\begin{gathered}:\longmapsto2(\text x + 6) = \text y \\ \end{gathered}

:⟼2(x+6)=y

\begin{gathered}:\longmapsto2\text x + 12 = \text y \\ \end{gathered}

:⟼2x+12=y

\begin{gathered}:\longmapsto2\bf x - y = - 12 \: ----(1) \\ \end{gathered}

:⟼2x−y=−12−−−−(1)

❒ When 7 is added to The Denominator :

Fraction Becomes : \dfrac{\text x}{\text y+6}

y+6

x

★ According To Question :

\begin{gathered} \dfrac{\text x}{\text y + 7} = \frac{1}{3} \\ \end{gathered}

y+7

x

=

3

1

\begin{gathered}:\longmapsto3\text x = \text y + 7 \\ \end{gathered}

:⟼3x=y+7

\begin{gathered}:\longmapsto \bf 3x - y = 7 \: - - - - (2) \\ \end{gathered}

:⟼3x−y=7−−−−(2)

✏ Subtracting (1) From (2) :

\begin{gathered}\purple{ \Large :\longmapsto \underline {\boxed{{\bf x = 19} }}} \\ \end{gathered}

:⟼

x=19

✏ Putting Value of x in (2) :

\begin{gathered}:\longmapsto3 \times 19 - \text y = 7 \\ \end{gathered}

:⟼3×19−y=7

\begin{gathered}:\longmapsto57 - \text y = 7 \\ \end{gathered}

:⟼57−y=7

\begin{gathered}:\longmapsto - \text y = 7 - 57 \\ \end{gathered}

:⟼−y=7−57

\begin{gathered}:\longmapsto \cancel - \text y = \cancel- 50 \\ \end{gathered}

:⟼

−

y=

−

50

\purple{ \Large :\longmapsto \underline {\boxed{{\bf y = 50} }}}:⟼

y=50

As,

Original Fraction = \dfrac{\text x}{\text y}

y

x

Hence,

\large\underline{\pink{\underline{\frak{\pmb{\text Original \:\: Fraction = \dfrac{19}{50} }}}}}

OriginalFraction=

50

19

OriginalFraction=

50

19