Math

QuestionTwo large and 1 small pump fill a pool in 4 hours. One large and 3 small pumps do the same. How long for 4 large and 4 small pumps?

Studdy Solution

STEP 1

Assumptions1. Two large and1 small pump can fill the swimming pool in4 hours. . One large and3 small pumps can also fill the swimming pool in4 hours.
3. We need to find out how many hours will it take4 large and4 small pumps to fill the swimming pool.
4. All large pumps are similar and all small pumps are also similar.

STEP 2

Let's denote the rate at which a large pump can fill the pool as L (pools/hour) and the rate for a small pump as (pools/hour). We can write the first condition as2+=142 + = \frac{1}{4}

STEP 3

The second condition can be written as+3=1 +3 = \frac{1}{}

STEP 4

We have a system of two equations. We can solve it by substitution or elimination. Let's subtract the second equation from the first to eliminate.
(2+)(+3)=1414(2 +) - ( +3) = \frac{1}{4} - \frac{1}{4}

STEP 5

implify the equation.
2=0 -2 =0

STEP 6

From this equation, we can see that the rate of a large pump is twice the rate of a small pump.
=2 =2

STEP 7

Substitute =2 =2 into the first equation.
2(2)+=142(2) + = \frac{1}{4}

STEP 8

implify the equation.
4+=144 + = \frac{1}{4}

STEP 9

olve for.
5=45 = \frac{}{4}=20 = \frac{}{20}

STEP 10

Substitute =20 = \frac{}{20} into the equation =2 =2 to find the rate of a large pump.
=2(20) =2(\frac{}{20})

STEP 11

olve for L.
=10 = \frac{}{10}

STEP 12

Now that we have the rates for both the large and small pumps, we can find out how long it will take4 large and4 small pumps to fill the pool. The total rate is4+4=Time4 +4 = Time

STEP 13

Substitute the values for L and into the equation.
(10)+(20)=Time(\frac{}{10}) +(\frac{}{20}) = Time

STEP 14

implify the equation.
410+420=Time\frac{4}{10} + \frac{4}{20} = Time

STEP 15

olve for Time.
Time=25+5=35Time = \frac{2}{5} + \frac{}{5} = \frac{3}{5}

STEP 16

The time is given in the form of a fraction of an hour. To convert it to hours, we take the reciprocal of the fraction.
Time=35Time = \frac{}{\frac{3}{5}}

STEP 17

olve for Time.
Time=53.67hoursTime = \frac{5}{3} \approx.67 \, hoursIt will take approximately.67 hours for4 large and4 small pumps to fill the swimming pool.

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