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Hint: Approach the solution by considering a fraction and proceed with further simplification by applying the given conditions.

Let us consider the fraction as $\dfrac{x}{y}$

By applying the ${1^{st}}$ condition we will get

$ \Rightarrow \dfrac{{x + 1}}{{y + 1}} = \dfrac{4}{5}$

On further simplification we will get

$\

\Rightarrow 5\left( {x + 1} \right) = 4\left( {y + 1} \right) \\

\Rightarrow 5x + 5 = 4y + 4 \\

\ $

$ \Rightarrow 5x - 4y = - 1$$ \to (1)$

And now by applying ${2^{nd}}$condition we will get

$\

\Rightarrow \dfrac{{x - 5}}{{y - 5}} = \dfrac{1}{2} \\

\\

\ $

On further simplification we will get

$\

\Rightarrow 2(x - 5) = 1(y - 5) \\

\Rightarrow 2x - 10 = y - 5 \\

\Rightarrow 2x - y = - 5 + 10 \\

\ $

$2x - y = 5 \to (2)$

For further calculation multiply equation $(2) \times 4$

$\

\Rightarrow 4 \times (2x - y = 5) \\

\Rightarrow 8x - 4y = 20 \to (3) \\

\ $

Now subtract equation $(3)$ from equation $(1)$ we get

$x = 7$

Putting $x = 7$ in equation $(2)$ we get

$\

\Rightarrow 2x - y = 5 \\

\Rightarrow 2(7) - y = 5 \\

\Rightarrow y = 9 \\

\ $

Here we got both $x\& y$ values

Therefore required fraction is $\dfrac{x}{y} = \dfrac{7}{9}$

Note: Apply the conditions in a step-by-step process with the proper approach to get the answer as the given problem is full of simplification.

Let us consider the fraction as $\dfrac{x}{y}$

By applying the ${1^{st}}$ condition we will get

$ \Rightarrow \dfrac{{x + 1}}{{y + 1}} = \dfrac{4}{5}$

On further simplification we will get

$\

\Rightarrow 5\left( {x + 1} \right) = 4\left( {y + 1} \right) \\

\Rightarrow 5x + 5 = 4y + 4 \\

\ $

$ \Rightarrow 5x - 4y = - 1$$ \to (1)$

And now by applying ${2^{nd}}$condition we will get

$\

\Rightarrow \dfrac{{x - 5}}{{y - 5}} = \dfrac{1}{2} \\

\\

\ $

On further simplification we will get

$\

\Rightarrow 2(x - 5) = 1(y - 5) \\

\Rightarrow 2x - 10 = y - 5 \\

\Rightarrow 2x - y = - 5 + 10 \\

\ $

$2x - y = 5 \to (2)$

For further calculation multiply equation $(2) \times 4$

$\

\Rightarrow 4 \times (2x - y = 5) \\

\Rightarrow 8x - 4y = 20 \to (3) \\

\ $

Now subtract equation $(3)$ from equation $(1)$ we get

$x = 7$

Putting $x = 7$ in equation $(2)$ we get

$\

\Rightarrow 2x - y = 5 \\

\Rightarrow 2(7) - y = 5 \\

\Rightarrow y = 9 \\

\ $

Here we got both $x\& y$ values

Therefore required fraction is $\dfrac{x}{y} = \dfrac{7}{9}$

Note: Apply the conditions in a step-by-step process with the proper approach to get the answer as the given problem is full of simplification.

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