Grade 12 math problem

Question

ceandayaTyro

Asked: April 17, 20212021-04-17T06:05:14+00:00 2021-04-17T06:05:14+00:00In: Arithmetic, Numbers and Combinatorics, High School/Senior Secondary School

Two large and 1 small pumps can fill a swimming pool in 4 hours. One large and 3 small pumps can also fill the same swimming pool in 4 hours. How many hours will it take 4 large and 4 small pumps to fill the swimming pool.(We assume that all large pumps are similar and all small pumps are also similar.)

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acedstud · Answer 1 · 2021-04-17T06:39:32+00:00

Let the time it will take 1 large pump to fill the pool be $x$ and the time it will take 1 small pump to fill the pool be $y$ , then

$\dfrac{2}{x}+\dfrac{1}{y}=\dfrac{1}{4}$ . Multiplying through by the LCD, $4xy$ , gives $8y+4x=xy$ . . . . . (1).

Also, $\dfrac{1}{x}+\dfrac{3}{y}=\dfrac{1}{4}$ . Multiplying through by the LCD, $4xy$ , gives $4y+12x=xy$ . . . . . (2).

Equating (1) and (2) gives $8y+4x=4y+12x\Rightarrow4y=8x\Rightarrow y=2x$ .

Substituting for $y$ in (1) gives $8(2x)+4x=x(2x)\Rightarrow16x+4x=2x^2\Rightarrow20x=2x^2\Rightarrow2x=20\Rightarrow x=10$ .

Thus, $y=2(10)=20$ .

Thus, it takes 1 large pump 10 hours to fill the pool and 1 small pump 20 hours to fill the pool.

Let the number of hours it will take 4 large and 4 small pumps to fill the swimming pool be $t$ , then

$\dfrac{4}{10}+\dfrac{4}{20}=\dfrac{1}{t}\Rightarrow\dfrac{8}{20}+\dfrac{4}{20}=\dfrac{1}{t}$ .

$\Rightarrow\dfrac{12}{20}=\dfrac{1}{t}\Rightarrow t=\dfrac{20}{12}=1\dfrac{2}{3}$ hours = 1 hour 40 minutes.

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