A fraction becomes 4.5 if 1 …

Let numerator of fraction is x and denominator is y.
Let the fraction be {tex}\frac { x } { y }.{/tex}
Then, according to the given conditions, we have
If 1 is added to both numerator and denominator then fraction becomes  {tex}\frac { 4 } { 5 }{/tex}.
{tex}\frac { x + 1 } { y + 1 } = \frac { 4 } { 5 } {/tex}
{tex}\Rightarrow{/tex} 5x +5 =4y + 4
{tex}\Rightarrow{/tex} 5x - 4y + 1=0 ...... (i)
If  5 is subtracted from both numerator and denominator, the fraction becomes {tex}\frac { 1 } {2}{/tex}.
{tex}\frac { x - 5 } { y - 5 } = \frac { 1 } { 2 }{/tex}
{tex}\Rightarrow{/tex} 2x - 10 = y - 5
{tex}\Rightarrow{/tex} 2x - y - 5=0 ........ (ii)
From (i) and (ii)
By using cross-multiplication, we have
{tex}\frac { x } { - 4 \times - 5 - ( - 1 ) \times 1 } = \frac { - y } { 5 \times - 5 - 2 \times 1 } = \frac { 1 } { 5 \times - 1 - 2 \times - 4 }{/tex}
{tex}\Rightarrow \quad \frac { x } { 20 + 1 } = \frac { y } { 25 + 2 } = \frac { 1 } { - 5 + 8 }{/tex}
{tex}\Rightarrow \quad \frac { x } { 21 } = \frac { y } { 27 } = \frac { 1 } { 3 } {/tex}
{tex}\Rightarrow x = \frac { 21 } { 3 } = 7 \text { and } y = \frac { 27 } { 3 } = 9{/tex}

Hence, the given fraction is 7/9.