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Question Number 44900 by peter frank last updated on 06/Oct/18

Answered by MJS last updated on 06/Oct/18

$$1\mathrm{large}\mathrm{pump}\mathrm{needs}x\mathrm{hours}\mathrm{to}\mathrm{fill}1\mathrm{pool}$$$$\mathrm{this}\mathrm{means}\mathrm{it}\mathrm{fills}\frac{1}{x}\mathrm{pool}\mathrm{per}\mathrm{hour}$$$$\mathrm{similar}1\mathrm{small}\mathrm{pump}\mathrm{fills}\frac{1}{y}\mathrm{pool}\mathrm{per}\mathrm{hour}$$$$2\mathrm{large}\mathrm{pumps}+1\mathrm{small}\mathrm{pump}\mathrm{need}4\mathrm{hours}$$$$\mathrm{to}\mathrm{fill}1\mathrm{pool}\Rightarrow \mathrm{they}\mathrm{fill}\frac{1}{4}\mathrm{pool}\mathrm{per}\mathrm{hour}$$$$\frac{2}{x}+\frac{1}{y}=\frac{1}{4}\Rightarrow 8y+4x=xy$$$$\mathrm{similar}:$$$$\frac{1}{x}+\frac{3}{y}=\frac{1}{4}\Rightarrow 4y+12x=xy$$$$8y+4x=4y+12x\Rightarrow 4y=8x\Rightarrow y=2x$$$$$$$$16x+4x=2x^2\Rightarrow 10x=x^2$$$$x\ne 0\Rightarrow \mathrm{divide}\mathrm{by}x$$$$x=10$$$$y=20$$$$\mathrm{this}\mathrm{means}1\mathrm{large}\mathrm{pump}\mathrm{fills}\frac{1}{10}\mathrm{pool}\mathrm{per}\mathrm{hour}$$$$1\mathrm{small}\mathrm{pump}\mathrm{fills}\frac{1}{20}\mathrm{pool}\mathrm{per}\mathrm{hour}$$$$\mathrm{together}\mathrm{they}\mathrm{fill}\frac{1}{10}+\frac{1}{20}=\frac{3}{20}\mathrm{pool}\mathrm{per}\mathrm{hour}$$$$\mathrm{each}4\mathrm{of}\mathrm{them}\mathrm{fill}\frac{12}{20}=\frac{3}{5}\mathrm{pool}\mathrm{per}\mathrm{hour}$$$$\Rightarrow \mathrm{they}\mathrm{need}\frac{5}{3}\mathrm{hours}=100\mathrm{minutes}\mathrm{to}\mathrm{fill}\mathrm{the}\mathrm{pool}$$

Commented by peter frank last updated on 06/Oct/18

$$\mathrm{thanks}\mathrm{sir}$$

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