If 5 men and 3 women can reap 18 acre of crop in 4 days, 3 men and 2 women can reap 22 acre of crop in 8 days, then how many men are required to join 21 women to reap 54 acre of crop in 6 days ?
Correct Answer: 5
Acreage reaped by 5 men and 3 women in 1 day
$$\eqalign{ & = \frac{{18}}{4} \cr & = \frac{9}{2} \cr} $$
Acreage reaped by 3 men and 2 women in 1 day
$$\eqalign{ & = \frac{{22}}{8} \cr & = \frac{{11}}{4} \cr} $$
Suppose 1 man can reap x acres in 1 day and 1 women can reap y acres in 1 day
$$\eqalign{ & \therefore 5x + 3y = \frac{9}{2} \cr & \Rightarrow 10x + 6y = 9\,.....{\text{(i)}} \cr & 3x + 2y = \frac{{11}}{4} \cr & \Rightarrow 9x + 6y = \frac{{33}}{4}\,.....{\text{(ii)}} \cr & {\text{Subtracting (ii) from (i),}} \cr & {\text{We get}}:x = 9 - \frac{{33}}{4} = \frac{3}{4} \cr & {\text{Putting x}} = \frac{3}{4}{\text{ in (i), we get}} \cr & \Rightarrow 6y = 9 - \frac{{15}}{2} \cr & \Rightarrow 6y = \frac{3}{2} \cr & \Rightarrow y = \frac{1}{4} \cr} $$
Acreage reaped by 21 women in 6 days
$$\eqalign{ & = \left( {\frac{1}{4} \times 21 \times 6} \right) \cr & = \frac{{63}}{2} \cr} $$
Remaining acreage to be reaped
$$\eqalign{ & = \left( {54 - \frac{{63}}{2}} \right) \cr & = \frac{{45}}{2} \cr} $$
Acreage reaped by 1 men in 6 days
$$\eqalign{ & = \left( {\frac{3}{4} \times 6} \right) \cr & = \frac{9}{2} \cr} $$
In 6 days, $$\frac{9}{2}$$ acre is reaped by 1 man
∴ In 6 days, $$\frac{{45}}{2}$$ acre is reaped by
$$\eqalign{ & = \left( {\frac{2}{9} \times \frac{{45}}{2}} \right){\text{men}} \cr & = 5{\text{ men}} \cr} $$
$$\eqalign{ & = \frac{{18}}{4} \cr & = \frac{9}{2} \cr} $$
Acreage reaped by 3 men and 2 women in 1 day
$$\eqalign{ & = \frac{{22}}{8} \cr & = \frac{{11}}{4} \cr} $$
Suppose 1 man can reap x acres in 1 day and 1 women can reap y acres in 1 day
$$\eqalign{ & \therefore 5x + 3y = \frac{9}{2} \cr & \Rightarrow 10x + 6y = 9\,.....{\text{(i)}} \cr & 3x + 2y = \frac{{11}}{4} \cr & \Rightarrow 9x + 6y = \frac{{33}}{4}\,.....{\text{(ii)}} \cr & {\text{Subtracting (ii) from (i),}} \cr & {\text{We get}}:x = 9 - \frac{{33}}{4} = \frac{3}{4} \cr & {\text{Putting x}} = \frac{3}{4}{\text{ in (i), we get}} \cr & \Rightarrow 6y = 9 - \frac{{15}}{2} \cr & \Rightarrow 6y = \frac{3}{2} \cr & \Rightarrow y = \frac{1}{4} \cr} $$
Acreage reaped by 21 women in 6 days
$$\eqalign{ & = \left( {\frac{1}{4} \times 21 \times 6} \right) \cr & = \frac{{63}}{2} \cr} $$
Remaining acreage to be reaped
$$\eqalign{ & = \left( {54 - \frac{{63}}{2}} \right) \cr & = \frac{{45}}{2} \cr} $$
Acreage reaped by 1 men in 6 days
$$\eqalign{ & = \left( {\frac{3}{4} \times 6} \right) \cr & = \frac{9}{2} \cr} $$
In 6 days, $$\frac{9}{2}$$ acre is reaped by 1 man
∴ In 6 days, $$\frac{{45}}{2}$$ acre is reaped by
$$\eqalign{ & = \left( {\frac{2}{9} \times \frac{{45}}{2}} \right){\text{men}} \cr & = 5{\text{ men}} \cr} $$