Question

Fraction becomes 4/5 of 1 is added to both numerator and denominator. If, however, 5 is subtracted from both numerator and denominator, fraction becomes 1/2. What is the fraction?

Answer 1

5x+5=4y+4.......................................(1)
2x-10=y-5.........................................(2)
 now after solving these eqns
x=7 & y=9
so the fraction is 7/9 

Answer 2

$\Large{\textbf{\underline{\underline{According\:to\:the\:Question}}}}$

Assumption

$\tt{\rightarrow Fraction=\dfrac{p}{c}}$

$\textbf{\underline{When\;1\;is\;added\;to\;Both}}$

${\boxed{\sf\:{\dfrac{p+1}{c+1}=\dfrac{4}{5}}}}$

5p + 5 = 4c + 4

5p - 4c = -1 ... (1)

$\textbf{\underline{When\;5\;is\;subtracted\;to\;Both}}$

${\boxed{\sf\:{\dfrac{p-5}{c-5}=\dfrac{1}{2}}}}$

2p - c = 5 ....(2)

$\textbf{\underline{Multiply\;(2)\;by\;5\;and\;(1)\;by\;2}}$

2(5p - 4c = -1)

10p - 8c = -2 .... (3)

5(2p - c = 5)

10p - 5c = 25 ..... (4)

$\textbf{\underline{Solving\;(3)\;and\;(4)}}$

c = 9

$\textbf{\underline{Put\;the\;value\;in\;(1)}}$

5p - 4c = -1

5p - 4(9) = -1

5p - 36 = -1

5p = -1 + 36

5p = 35

$\tt{\rightarrow p=\dfrac{35}{5}}$

p = 7

$\textbf{\underline{Hence\;we\;get}}$

p = 7 and c = 9

$\Large{\boxed{\sf\:{Fraction=\dfrac{7}{9}}}}$