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Mathematics

6 men and 8 boys can finish a piece of work in 14 days while 8 men and 12 boys can do it in 10 days. Find the time taken by one man alone and by one boy alone to finish the work.

Linear Equations

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Answer

Lets assume that one man takes x days to do work and y days for one boy.

So, the amount of work done by 1 man in 1 day = 1x\dfrac{1}{x}

So, the amount of work done by 1 boy in 1 day = 1y\dfrac{1}{y}

Given,

6 men and 8 boys finish the work in 14 days,

⇒6x+8y=114 ….(1) \Rightarrow \dfrac{6}{x} + \dfrac{8}{y} = \dfrac{1}{14} \text{ ….(1) }

Given,

8 men and 12 boys finish the same work in 10 days,

⇒8x+12y=110\Rightarrow \dfrac{8}{x} + \dfrac{12}{y} = \dfrac{1}{10}     ….(2)

Multiply equation (1) by 2, we get:

⇒2(6x+8y)=114×2⇒12x+16y=17 ….(3) \Rightarrow 2 \Big(\dfrac{6}{x} + \dfrac{8}{y}\Big) = \dfrac{1}{14} \times 2 \\[1em] \Rightarrow \dfrac{12}{x} + \dfrac{16}{y} = \dfrac{1}{7} \text{ ….(3) }

Multiply equation (2) by 32\dfrac{3}{2}, we get:

⇒32(8x+12y)=110×32⇒12x+18y=320 ….(4) \Rightarrow \dfrac{3}{2} \Big(\dfrac{8}{x} + \dfrac{12}{y}\Big) = \dfrac{1}{10} \times \dfrac{3}{2} \\[1em] \Rightarrow \dfrac{12}{x} + \dfrac{18}{y}= \dfrac{3}{20} \text{ ….(4) }

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

⇒(12x+18y)−(12x+16y)=320−17⇒(12x+18y−12x−16y)=21140−20140⇒18y−16y=1140⇒2y=1140⇒y=2×140=280.\Rightarrow \Big(\dfrac{12}{x} + \dfrac{18}{y}\Big) - \Big(\dfrac{12}{x} + \dfrac{16}{y}\Big) = \dfrac{3}{20} - \dfrac{1}{7} \\[1em] \Rightarrow \Big(\dfrac{12}{x} + \dfrac{18}{y} - \dfrac{12}{x} - \dfrac{16}{y}\Big) = \dfrac{21}{140} - \dfrac{20}{140} \\[1em] \Rightarrow \dfrac{18}{y} - \dfrac{16}{y} = \dfrac{1}{140} \\[1em] \Rightarrow \dfrac{2}{y} = \dfrac{1}{140} \\[1em] \Rightarrow y = 2 \times 140 = 280.

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

⇒6x+8y=114⇒6x+8280=114⇒6x=114−8280⇒6x=20280−8280⇒6x=12280⇒6×28012=x⇒x=140.\Rightarrow \dfrac{6}{x} + \dfrac{8}{y} = \dfrac{1}{14} \\[1em] \Rightarrow \dfrac{6}{x} + \dfrac{8}{280} = \dfrac{1}{14} \\[1em] \Rightarrow \dfrac{6}{x} = \dfrac{1}{14} - \dfrac{8}{280} \\[1em] \Rightarrow \dfrac{6}{x} = \dfrac{20}{280} - \dfrac{8}{280} \\[1em] \Rightarrow \dfrac{6}{x} = \dfrac{12}{280} \\[1em] \Rightarrow \dfrac{6 \times 280}{12} = x \\[1em] \Rightarrow x = 140.

Hence, one man can finish the work in = 140 days and one boy finish the work in = 280 days .

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