Question

$2$ women and $5$ men can together finish an embroidery work in $4$ days, while $3$ women and $6$ men can finish it in $3$ days. Find the time taken by $1$ woman as well as $1$ man to finish the work if each of them works alone.

• A. Time taken by $1$ woman alone to finish the work: $24$ days, and also that taken by $1$ man alone: $32$ days
• B. Time taken by $1$ woman alone to finish the work: $18$ days, and also that taken by $1$ man alone: $36$ days
• C. Time taken by $1$ woman alone to finish the work: $14$ days, and also that taken by $1$ man alone: $30$ days
• D. Time taken by $1$ woman alone to finish the work: $12$ days, and also that taken by $1$ man alone: $28$ days

Answer 1

Answer: (B)

Time taken by $1$ woman alone to finish the work: $18$ days, and also that taken by $1$ man alone: $36$ days

Let the work done by man and woman per day be $x$ and $y$ respectively.

When the work is completed in $4$ days

Since $5$ men and $2$ women complete the work in $4$ days

therefore work done by $5$ men and $2$ women in $1$ day $=$ $\frac{1}{4}$

$\therefore 5x+2y=\frac{1}{4}⟶{eq}^{n}1$

When the work is completed in $3$ days

Since $6$ men and $3$ women complete the work in $3$ days

therefore work done by $6$ men and $3$ women in $1$ day $=$ $\frac{1}{3}$

$\therefore 6x+3y=\frac{1}{3}⟶{eq}^{n}2$

Multiplying by $3$ in ${eq}^{n}1$, we get

$\Rightarrow 15x+6y=\frac{3}{4}⟶{eq}^{n}3$

Multiplying by $2$ in ${eq}^{n}2$, we get

$\Rightarrow 12x+6y=\frac{2}{3}⟶{eq}^{n}4$

On subtracting ${eq}^n$ 4 from ${eq}^n$ 3, we get

$\Rightarrow 15x+6y-12x-6y=\frac{3}{4}-\frac{2}{3}$

$\Rightarrow 3x=\frac{1}{12}$

$\Rightarrow x=\frac{1}{36}$

On substituting the value of $x$ in ${eq}^{n}2$, we get

$\Rightarrow 6\times \frac{1}{36}+3y=\frac{1}{3}$

$\Rightarrow 3y=\frac{1}{3}-\frac{1}{6}$

$\Rightarrow y=\frac{1}{18}$

Thus,

work done by $1$ man in $1$ day $=\frac{1}{36}$ days

$\therefore$ Time taken by $1$ man alone to finish the work $=36$ days

work done by $1$ woman in $1$ day $=\frac{1}{18}$ days

$\therefore$ Time taken by $1$ woman alone to finish the work $=18$ days