[50 days] A garrison had provisions for a certain number of days. After 10 days \dfrac{1}{5} of the men dese

For Competitive Exams

A garrison had provisions for a certain number of days. After 10 days $\dfrac{1}{5}$ of the men desert and it is found that the provisions will now last just as long as before. How long was that?

50 days

30 days

40 days

60 days

Explanation:

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

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

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

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

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

$\Rightarrow y = 50$

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