Question

In how many days can $10$ women finish a work?
$I$. $10$ men can complete the work in $6$ days.
$II$. $10$ men and $10$ women together can complete the work in $3\frac{3}{7}$ days.
$III$. If $10$ men work for $3$ days and thereafter $10$ women replace them, the remaining work in completed in $4$ days.

• A. Any two of the three
• B. $I$ and $II$ only
• C. $II$ and $III$ only
• D. $I$ and $III$ only
• E. None of these

Answer 1

The correct option is A Any two of the three

$I$. $(10\times 6)$ men can complete the work in $1$ day.
$\Rightarrow$ $1$ man's $1$ day's work $=\frac{1}{60}$
$II$. $(10\times \frac{24}{7})men+(10\times \frac{24}{7})women$ can complete the work in $1$ day.
$\Rightarrow$ $\left(\frac{240}{7}\right)$ men's $1$ day work $+\left(\frac{240}{7}\right)$ women's $1$ day work $=1$
$\Rightarrow$ $(\frac{240}{7}\times \frac{1}{60})+\left(\frac{240}{7}\right)$ women's $1$ day Work $=1$.
$\Rightarrow$ $\left(\frac{240}{7}\right)$ women's $1$ day work $=(1-\frac{4}{7})=\frac{3}{7}$
$\Rightarrow$ $10$ women's $1$ day's work $=(\frac{3}{7}\times \frac{240}{7}\times 10)=\frac{1}{8}$
So, $10$ women can finish the work in $8$ days.
$III$. ($10$ men'w work for $3$ days)$+$ ($10$ women's work for $4$ days)$=1$
$\Rightarrow$ $(10\times 3)$ men's $1$ day's work $+$ $(10\times 4)$ women's $1$ day's work $=1$
$\Rightarrow$ $30$ men's $1$ day's work $+$ $40$ women's $1$ day's work $=1$
Thus, $I$ and $III$ will give us the answer.
And, $II$ and $III$ will give us the answer.
$\therefore$ Correct answer is (A).