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Answers
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