The work done by a women in 8 hours is equal to the work done by a man in 6 hours and by a boy in 12 hours. If working 6 hours per day 9 men can complete a work in 6 days, then in how many days can 12 men, 12 women and 12 boys together finish the same work, working 8 hours per day ?

1 $1 \frac{1}{2} days$ ${\text{1}}\frac{1}{2}{\text{ days}}$
2 $3 days$ ${\text{3 days}}$
3 $3 \frac{2}{3} days$ ${\text{3}}\frac{2}{3}{\text{ days}}$
4 $4 \frac{1}{2} days$ ${\text{4}}\frac{1}{2}{\text{ days}}$

Answer:- 1

Explanation:-

Solution:

Ratio of time taken by a woman, a man and a boy

\begin{aligned} = 8 : 6 : 12 \\ = 4 : 3 : 6 \end{aligned}

$\eqalign{ & = 8:6:12 \cr & = 4:3:6 \cr}$
So, 4 women ≡ 3 men ≡ 6 boy
(12 mens + 12 womens + 12 boys)

\begin{aligned} = [12 + (\frac{3}{4} \times 12) + (\frac{3}{6} \times 12)] men \\ = (12 + 9 + 6) men \\ = 27 men \end{aligned}

$\eqalign{ & = \left[ {12 + \left( {\frac{3}{4} \times 12} \right) + \left( {\frac{3}{6} \times 12} \right)} \right]{\text{men}} \cr & {\text{ = }}\left( {12 + 9 + 6} \right){\text{men}} \cr & = 27{\text{ men}} \cr}$
Let the required number of days be x
More men, Less days (Indirect proportion)
More working hours, Less days (Indirect proportion)

\begin{matrix} Working hours 8 : 6 \\ Men 27 : 9 \end{matrix}} :: 6 : x

$\left. \begin{gathered} {\text{Working hours 8}}:6 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Men 27}}:9 \hfill \\ \end{gathered} \right\}::6:x$

\begin{aligned} ∴ 27 \times 8 \times x = 9 \times 6 \times 6 \\ \Leftrightarrow x = \frac{(9 \times 6 \times 6)}{(27 \times 8)} \\ \Leftrightarrow x = \frac{3}{2} \\ \Leftrightarrow x = 1 \frac{1}{2} \end{aligned}

$\eqalign{ & \therefore \,27 \times 8 \times x = 9 \times 6 \times 6 \cr & \Leftrightarrow x = \frac{{\left( {9 \times 6 \times 6} \right)}}{{\left( {27 \times 8} \right)}} \cr & \Leftrightarrow x = \frac{3}{2} \cr & \Leftrightarrow x = 1\frac{1}{2} \cr}$