Question

8 girls and 12 boys can finish the work in 10 days while 6 girls and 8 boys can finish it in 14 days.find the time taken by one girl alone and then one boy alone to finish the work.​
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Answer 1

Answer :-

Basic Concept :-

Representing Systems of Linear Equations from Word Problem -

Understand the problem.

• Understand all the words used in stating the problem.

• Understand what you are asked to find.

Translate the problem to an equation.

• Assign a variable (or variables) to represent the unknown.

• Clearly state what the variable represents.

Carry out the plan and solve the problem.

Let's do it now!!

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

$\begin{gathered}\begin{gathered}\bf \:Let - \begin{cases} &\sf{time \: tak \: en \: by \: girl \: be \: x \: days} \\ &\sf{time \: tak \: en \: by \: boy \: be \: y \: days} \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf \:Hence - \begin{cases} &\sf{girl \: 1 \: day \: work \: = \dfrac{1}{x} } \\ &\sf{boy \: 1 \: day \: work \: = \dfrac{1}{y} } \end{cases}\end{gathered}\end{gathered}$

$\large \underline{\tt \:{ According  \: to  \: statement }}$

• 8 girl and 12 boy can finished the work in 10 days

So,

$\rm :\longmapsto\:8 \: girl\: 1 \: day \: work \: = \dfrac{8}{x}$

and

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

So,

• Total one day work is given by

$\rm :\longmapsto\: \boxed{ \bf \: \dfrac{8}{x} + \dfrac{12}{y} = \dfrac{1}{10}} - - - (1)$

Again,

$\large \underline{\tt \:{ According  \: to  \: statement }}$

• 6 girl and 8 boy can finished it in 14 days.

So,

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

and

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

So,

• Total 1 day work is given by

$\rm :\longmapsto\: \boxed{ \bf \: \dfrac{6}{x} + \dfrac{8}{y} = \dfrac{1}{14}} - - - (2)$

Now,

Solve equation (1) and (2), to get the values of x and y

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

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

and

$\rm :\longmapsto\:\dfrac{24}{x} + \dfrac{32}{4} = \dfrac{4}{14}$

$\rm :\implies\:\dfrac{24}{x} + \dfrac{32}{y} = \dfrac{2}{7} - - - (4)$

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

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

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

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

$\bf\implies \:y = 280 \: days$

Substituting the value of y in equation (1), we get

$\rm :\longmapsto\:\dfrac{8}{x} + \dfrac{12}{280} = \dfrac{1}{10}$

$\rm :\longmapsto\:\dfrac{8}{x} + \dfrac{3}{70} = \dfrac{1}{10}$

$\rm :\longmapsto\:\dfrac{8}{x} = \dfrac{1}{10} - \dfrac{3}{70}$

$\rm :\longmapsto\:\dfrac{8}{x} = \dfrac{7 - 3}{70}$

$\rm :\longmapsto\:\dfrac{8}{x} = \dfrac{4}{70}$

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

$\bf\implies \:x = 140 \: days$

$\begin{gathered}\begin{gathered}\bf \:Hence- \begin{cases} &\sf{time \: tak \: en \: by \: girl \: \: = \: 140 \: days} \\ &\sf{time \: tak \: en \: by \: boy \: \: = \: 280 \: days} \end{cases}\end{gathered}\end{gathered}$

Answer 2

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$\huge\color{red}\boxed{\colorbox{aqua}{αηsωєя࿐꧂ }}\downarrow$

8 boys and 12 girls can finish the work in 10 days, while 6 boys and 8 girls can finish it in 14 days

Let x and y represent work done per day by boy and girl respectively:

8x + 12y = 1/10     

 6x  + 8y = 1/14    (multiplying by -3/2 to  eliminate the y variable)

8x + 12y = 1/10     

-9x  -12y = -3/28    (result of multiplying thru by -3/2 to  eliminate the y variable)

x = 1/140  and y = 1/280, work PER Day by each

x = 1/140  and y = 1/280, work PER Day by eachtime taken by 1 boy alone140 days and 1 girl alone 280 days to finish the work.

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