Question

If 2men and 3boys can do a piece of work in 10days and 3men and 2boys can do a piece of work in 8days. How long does 2men and 1boy take to do it
A) 15
B) 18
C) 12.5
D) 16

Answer 1

Hint:
Form an equation of each condition and solve for 1day work of boy and man. Use the condition (How long does 2 men and 1 boy take to do it) to solve the number of days it took for them.

Complete step by step solution:
Given, 2men and 3boys can do a piece of work in 10days.
3men and 2boys can do a piece of work in 8days.
Let 2men and 3boys 1day work be $\dfrac{1}{{10}}$ and 3men and 2boys 1day work be $\dfrac{1}{8}$.
Frame an equation with the above conditions.
$
  2m + 3b = \dfrac{1}{{10}}.........\left( 1 \right) \\
  3m + 2b = \dfrac{1}{8}...........\left( 2 \right) \\
 $
Solve for $m$and $b$, multiply 3 with eq (1) and 2 with eq (2)
$
   \Rightarrow 3\left( {2m + 3b = \dfrac{1}{{10}}} \right) \\
   \Rightarrow 2\left( {3m + 2b = \dfrac{1}{8}} \right) \\
 $
Solve for the 1day work of the boy.
$
   \Rightarrow 6m + 9b = \dfrac{3}{{10}} \\
   \Rightarrow 6m + 4b = \dfrac{2}{8} \\
 $
On solving we get,
$
   \Rightarrow 5b = \dfrac{3}{{10}} - \dfrac{1}{4} \\
    \\
 $

Take LCM, LCM is 20
$
   \Rightarrow 5b = \dfrac{3}{{10}} - \dfrac{1}{4} \\
   \Rightarrow 5b = \dfrac{6}{{20}} - \dfrac{5}{{20}} \\
   \Rightarrow 5b = \dfrac{1}{{20}} \\
   \Rightarrow b = \dfrac{1}{{100}}........\left( 3 \right) \\
 $
Substitute the value of (3) in equation (1), we get
\[
   \Rightarrow 2m + 3b = \dfrac{1}{{10}} \\
   \Rightarrow 2m + 3\left( {\dfrac{1}{{100}}} \right) = \dfrac{1}{{10}} \\
   \Rightarrow 2m = \dfrac{1}{{10}} - 3\left( {\dfrac{1}{{100}}} \right) \\
   \Rightarrow 2m = \dfrac{7}{{100}} \\
   \Rightarrow m = \dfrac{7}{{200}}.........\left( 4 \right) \\
 \]
Substitute the value of (1) and (2), for 1day work of 2men and 1boy.
$ \Rightarrow 2m + 1b = 2\left( {\dfrac{7}{{200}}} \right) + 1\left( {\dfrac{1}{{100}}} \right)$
Solve for 1day work and the reciprocal of the value is the days the combination can work.
$
 \Rightarrow 2m + 1b = 2\left( {\dfrac{7}{{200}}} \right) + 1\left( {\dfrac{1}{{100}}} \right) \\
= \dfrac{8}{{100}} \\
= \dfrac{2}{{25}} \\
 $

So, 2 men and 1 boy can do the work for 12.5 days.

Note:
Reciprocal work done by a particular combination or a person is 1 days work of the whole and vice-versa. This is an exam loving topic. you’ll always find questions from this topic in competitive exams.