Question

A fraction becomes $\frac{4}{5}$ if one is added to both numerator and denominator. If, however, 5 is subtracted from numerator and denominator, the resultant fraction becomes $\frac{1}{2}$. Find the fraction. [3 Marks]

Answer 1

Let x be the numerator and y be the denominator.
Then, it is given that $\frac{(x+1)}{(y+1)}=\frac{4}{5};\frac{(x-5)}{(y-5)}=\frac{1}{2}$
$\frac{(x+1)}{(y+1)}=\frac{4}{5}\Rightarrow 5x+5=4y+4\Rightarrow 5x-4y+1=0\dots .\left(1\right)$ [1 Mark]
$\frac{(x-5)}{(y-5)}=\frac{1}{2}\Rightarrow 2x-10=y-5\Rightarrow 2x-y-5=0\dots .\left(2\right)$
Multiplying (2) by 4, we get
8x - 4y - 20 = 0 …(3)
Subtracting (3) from (1), we get
-3x + 21 = 0
$\Rightarrow x=7$ [1 Mark]
Putting x = 7 in (2), we get $2\times 7-y-5$ = 0
$\Rightarrow y=9$.
Therefore, the fraction is $\frac{7}{9}$. [1 Mark]