Assume that initially garrison had provisions for $x$ men for $y$ days.

So, after 10 days, garrison had provisions for $x$ men for $(y-10)$ days

Also, after 10 days, garrison had provisions for $\dfrac{4x}{5}$ men for $y$ days $\left( ? x - \dfrac{x}{5} = \dfrac{4x}{5}\right)$

More men, Less days (Indirect Proportion)

(men) $x$ : $\dfrac{4x}{5}$ :: $y$ : $(y-10)$

$\Rightarrow x(y - 10) = \dfrac{4xy}{5}\\~\\$

$\Rightarrow (y - 10) = \dfrac{4y}{5}\\~\\$

$\Rightarrow 5(y - 10) = 4y\\~\\$

$\Rightarrow 5y - 50 = 4y \\~\\$

$\Rightarrow y = 50$