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Mathematics

B takes 16 days less than A to do a certain piece of work. If both working together can complete the work in 15 days, in how many days will B alone complete the work ?

Linear Equations

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Answer

Let A take x days to complete the work so B will take (x - 16) days.

Work done by A alone in 1 day = 1x\dfrac{1}{x}

Work done by B alone in 1 day = 1x−16\dfrac{1}{x - 16}

Work done by A and B together in 1 day is 1x+1x−16\dfrac{1}{x} + \dfrac{1}{x - 16}

Given,

If both work together, they finish the work in 15 days.

∴15(1x+1x−16)=1⇒x−16+xx(x−16)=115⇒2x−16x2−16x=115⇒15(2x−16)=x2−16x⇒30x−240=x2−16x⇒x2−16x−30x+240=0⇒x2−46x+240=0⇒x2−40x−6x+240=0⇒x(x−40)−6(x−40)=0⇒(x−6)(x−40)=0⇒x−6=0 or x−40=0⇒x=6 or x=40.\therefore 15\Big(\dfrac{1}{x} + \dfrac{1}{x - 16}\Big) = 1 \\[1em] \Rightarrow \dfrac{x - 16 + x}{x(x - 16)} = \dfrac{1}{15} \\[1em] \Rightarrow \dfrac{2x - 16}{x^2 - 16x} = \dfrac{1}{15} \\[1em] \Rightarrow 15(2x - 16) = x^2 - 16x \\[1em] \Rightarrow 30x - 240 = x^2 - 16x \\[1em] \Rightarrow x^2 - 16x - 30x + 240 = 0 \\[1em] \Rightarrow x^2 - 46x + 240 = 0 \\[1em] \Rightarrow x^2 - 40x - 6x + 240 = 0 \\[1em] \Rightarrow x(x - 40) - 6(x - 40) = 0 \\[1em] \Rightarrow (x - 6)(x - 40) = 0 \\[1em] \Rightarrow x - 6 = 0 \text{ or } x - 40 = 0 \\[1em] \Rightarrow x = 6 \text{ or } x = 40.

Value of x will be greater than 16.

∴ x = 40 and x - 16 = 40 - 16 = 24.

Hence, B alone can finish the work in 24 days.

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