Question

A piece of work can be done by $2$ men and $7$ boys in $4$ days. The same piece of work can be done by $4$ men and $4$ boys in $3$ days. How long it would take to do the same work by one man or one boy?

Answer 1

Let $x$ be the number of days in which one man can complete the work.
Let y be the number of days in which one boy can complete the work.
In one day a man can do $\frac{1}{x}th$ work and a boy does $\frac{1}{y}th$ work.
Now $2$ men and $7$ boys can complete the work in $4$ days. Hence they can complete $\frac{1}{4}$th of the work in one day. Thus we obtain
$\frac{2}{x}+\frac{7}{y}=\frac{1}{4}$ ..............($1$).
Again $4$ men and $4$ boys together complete the work in $3$ days. Hence they can complete $\frac{1}{3}rd$ of the work in a day. Thus we also get
$\frac{4}{x}+\frac{4}{y}=\frac{1}{3}.........\left(2\right)$
Multiply $\left(1\right)$ by $2$ and subtract $\left(2\right)$ from it, you get
$\frac{4}{x}+\frac{14}{y}-\frac{4}{x}-\frac{4}{y}=\frac{2}{4}-\frac{1}{3}$.
This reduces to $\frac{10}{y}=\frac{1}{6}$. Hence $y=60$. From equation $\left(2\right)$ we get
$\frac{4}{x}=\frac{1}{3}-\frac{4}{y}=\frac{1}{3}-\frac{4}{60}=\frac{1}{3}-\frac{1}{15}=\frac{4}{15}$.
Hence $x=15$.
Thus, one man can complete the work in $15$ days and one boy can do the work in $60$ days.