Question

$8$ men and $12$ boys can finish a piece of work in $10$ days, while $6$ men and $8$ boys can finish it in $14$ days. Find the time taken by one man alone and that by one boy alone to finish that work.

• A. Man: $160$ days. Boy : $190$ days
• B. Man: $140$ days. Boy : $280$ days
• C. Man: $240$ days. Boy : $150$days
• D. Man: $260$ days. Boy : $200$ days

Answer 1

The correct option is B Man: $140$ days. Boy : $280$ days

From Question:
Let one man alone can finish the work in $x$ days and one boy alone can finish the work in $y$ days.
In one day man can do $\frac{1}{x}th$ work , then 8 man can do $\frac{8}{x}th$ part of work.
In one day man can do $\frac{1}{y}th$ work , then 8 man can do $\frac{12}{y}th$ part of work.
Given together they can finish in 10 days
$\therefore \frac{8}{x}+\frac{12}{y}=\frac{1}{10}$
Again 6 man can do in one day = $\frac{6}{x}$ parts of the work.
Again 8 boy can do in one day = $\frac{8}{y}$ parts of the work.
Given that together they can finish in 14 days.
$\therefore \frac{6}{x}+\frac{8}{y}=\frac{1}{14}$
Putting $\frac{1}{x}=u$ and $\frac{1}{y}=v$ we get the following equations
$8u+12v=\frac{1}{10}$
$\Rightarrow 80u+120v=1$
And
$6u+8v=\frac{1}{14}$
$\Rightarrow 84u+112v=1$
By Cross multiplication, we get
$\Rightarrow \frac{u}{120-112}=\frac{v}{84-80}=\frac{-1}{8960-10080}$
$\Rightarrow \frac{u}{8}=\frac{v}{4}=\frac{-1}{-1120}$
$\Rightarrow u=\frac{-8}{-1120}=\frac{1}{140}$
$\Rightarrow v=\frac{-4}{-1120}=\frac{1}{280}$
Now $u=\frac{1}{140}$ $=\frac{1}{x}$ $\Rightarrow x=140$
Now $v=\frac{1}{280}$ $=\frac{1}{y}$ $\Rightarrow y=280$
Thus one man alone can finish work in $140$ days and one boy alone can finish work in $280$ days