Question

6 boys and 8 girls can finish a piece of work in 14 days while 8 boys and 12 girls can finish a piece of work in 10 days . Find the time taken by one boy alone and one girl alone to finish the work ?​

Answer 1

$\large\underline{\sf{Solution-}}$

Let assume that

Number of days taken by 1 boy alone to finish the work be x days and Number of days taken by 1 girl alone to finish the work be y days.

So, it implies,

$\rm :\longmapsto\:1 \: day \: work \: of \: 1 \: boy \: = \: \dfrac{1}{x}$

$\rm :\longmapsto\:1 \: day \: work \: of \: 1 \: girl \: = \: \dfrac{1}{y}$

According to statement,

6 boys and 8 girls can finish a piece of work in 14 days.

So,

$\rm :\longmapsto\:1 \: day \: work \: of \: 6 \: boys\: = \: \dfrac{6}{x}$

and

$\rm :\longmapsto\:1 \: day \: work \: of \: 8 \: girls\: = \: \dfrac{8}{y}$

$\rm :\implies\:\dfrac{6}{x} + \dfrac{8}{y} = \dfrac{1}{14} - - - - (1)$

According to statement again,

8 boys and 12 girls can finish a piece of work in 10 days.

$\rm :\longmapsto\:1 \: day \: work \: of \: 6 \: boys\: = \: \dfrac{8}{x}$

and

$\rm :\longmapsto\:1 \: day \: work \: of \: 12 \: girls\: = \: \dfrac{12}{y}$

So,

$\rm :\implies\:\dfrac{8}{x} + \dfrac{12}{y} = \dfrac{1}{10} - - - - (2)$

Now, multiply equation (1) by 4 and equation (2) by 3, we get

$\rm :\implies\:\dfrac{24}{x} + \dfrac{32}{y} = \dfrac{2}{7} - - - - (3)$

and

$\rm :\implies\:\dfrac{24}{x} + \dfrac{36}{y} = \dfrac{3}{10} - - - - (4)$

On Subtracting equation (3) from (4), we get

$\rm :\longmapsto\:\dfrac{4}{y} = \dfrac{3}{10} - \dfrac{2}{7}$

$\rm :\longmapsto\:\dfrac{4}{y} = \dfrac{21 - 20}{70}$

$\rm :\longmapsto\:\dfrac{4}{y} = \dfrac{1}{70}$

$\bf\implies \:y = 280$

On Substituting y = 280 in equation (1), we get

$\rm :\implies\:\dfrac{6}{x} + \dfrac{8}{280} = \dfrac{1}{14}$

$\rm :\longmapsto\:\:\dfrac{6}{x} + \dfrac{1}{35} = \dfrac{1}{14}$

$\rm :\longmapsto\:\:\dfrac{6}{x} = \dfrac{1}{14} - \dfrac{1}{35}$

$\rm :\longmapsto\:\:\dfrac{6}{x} = \dfrac{5 - 2}{70}$

$\rm :\longmapsto\:\:\dfrac{6}{x} = \dfrac{3}{70}$

$\rm :\longmapsto\:\:\dfrac{2}{x} = \dfrac{1}{70}$

$\bf\implies \:x = 140$

Hence,

Number of days taken by 1 boy alone to finish the work be 140 days

and

Number of days taken by 1 girl alone to finish the work be 280 days.