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 ?
Correct Answer: $${\text{1}}\frac{1}{2}{\text{ days}}$$
Ratio of time taken by a woman, a man and a boy
$$\eqalign{ & = 8:6:12 \cr & = 4:3:6 \cr} $$
So, 4 women ≡ 3 men ≡ 6 boy
(12 mens + 12 womens + 12 boys)
$$\eqalign{ & = \left{\text{men}} \cr & {\text{ = }}\left( {12 + 9 + 6} \right){\text{men}} \cr & = 27{\text{ men}} \cr} $$
Let the required number of days be x
More men, Less days (Indirect proportion)
More working hours, Less days (Indirect proportion)
\
$$\eqalign{ & \therefore \,27 \times 8 \times x = 9 \times 6 \times 6 \cr & \Leftrightarrow x = \frac{{\left( {9 \times 6 \times 6} \right)}}{{\left( {27 \times 8} \right)}} \cr & \Leftrightarrow x = \frac{3}{2} \cr & \Leftrightarrow x = 1\frac{1}{2} \cr} $$
$$\eqalign{ & = 8:6:12 \cr & = 4:3:6 \cr} $$
So, 4 women ≡ 3 men ≡ 6 boy
(12 mens + 12 womens + 12 boys)
$$\eqalign{ & = \left{\text{men}} \cr & {\text{ = }}\left( {12 + 9 + 6} \right){\text{men}} \cr & = 27{\text{ men}} \cr} $$
Let the required number of days be x
More men, Less days (Indirect proportion)
More working hours, Less days (Indirect proportion)
\
$$\eqalign{ & \therefore \,27 \times 8 \times x = 9 \times 6 \times 6 \cr & \Leftrightarrow x = \frac{{\left( {9 \times 6 \times 6} \right)}}{{\left( {27 \times 8} \right)}} \cr & \Leftrightarrow x = \frac{3}{2} \cr & \Leftrightarrow x = 1\frac{1}{2} \cr} $$