Question

$2$ men and $3$ boys together can do a piece of work in $8$ days. The same work is done in $6$ days by $3$ men and $2$ boys together. How long would $1$ boy alone or $1$ man alone take to complete the work.

• A. One boy alone can do a piece of work in $130$ days
  and one man alone can do a piece of work in $160$ days
• B. One boy alone can do a piece of work in $90$ days
  and one man alone can do a piece of work in $70$ days
• C. One boy alone can do a piece of work in $120$ days
  and one man alone can do a piece of work in $20$ days
• D. One boy alone can do a piece of work in $160$ days
  and one man alone can do a piece of work in $12$ days

Answer 1

Answer: (C)

One boy alone can do a piece of work in $120$ days

and one man alone can do a piece of work in $20$ days
Let $1$ boy alone can do a piece of work in $x$ days and $1$ man alone can do a piece of work in $y$ days.
So, $1$ day work of $1$ boy $=\frac{1}{x}$
and $1$ day work of $1$ man $=\frac{1}{y}$
ATP
$\frac{3}{x}+\frac{2}{y}=\frac{1}{8}$ .........(i)
and $\frac{2}{x}+\frac{3}{y}=\frac{1}{6}$ .......(ii)
Multiplying equation (i) by $2$ and equation (ii) by $3$
$\frac{6}{x}+\frac{4}{y}=\frac{1}{4}$
$\frac{6}{x}+\frac{9}{y}=\frac{1}{2}$
------------------------------------------
$-\frac{5}{y}=-\frac{1}{4}$
$\Rightarrow$ $y=20$
substitute $y=20$ in equation (i)
$\frac{3}{x}+\frac{3}{20}=\frac{1}{8}$
$\frac{3}{x}=\frac{1}{8}-\frac{1}{10}$
$\frac{3}{x}=\frac{5-4}{40}$
$\frac{3}{x}=\frac{1}{40}$
$\Rightarrow$ $x=120$
So, one boy alone can do a piece of work in $120$ days
and one man alone can do a piece of work in $20$ days.