Pumps P_1 and P_2 operating together can fill a water storage tank in 12\text{ hours}, whi… — JAMB UTME

Pumps $P_1$ and $P_2$ operating together can fill a water storage tank in $12\text{ hours}$ , while pumps $P_2$ and $P_3$ operating together can fill the same tank in $20\text{ hours}$ . When all three pumps operate simultaneously for $5\text{ hours}$ , they fill exactly $\frac{1}{2}$ of the tank. If pump $P_2$ is then shut off, how many additional hours will it take pumps $P_1$ and $P_3$ working together to fill the remainder of the tank?

$7.5\text{ hours}$ Cevap
B
$15.0\text{ hours}$
C
$6.0\text{ hours}$
D
$12.5\text{ hours}$

Cevap

7.5\text{ hours}

The three pumps combined have a work rate of

\frac{1}{10}\text{ tank/hr}

. Subtracting the rate of

P_1 + P_2

(which is

\frac{1}{12}

) gives

P_3

's rate as

\frac{1}{60}\text{ tank/hr}

. Subtracting the rate of

P_2 + P_3

(which is

\frac{1}{20}

) gives

P_1

's rate as

\frac{1}{20}\text{ tank/hr}

. Together,

P_1

and

P_3

have a combined rate of

\frac{1}{20} + \frac{1}{60} = \frac{1}{15}\text{ tank/hr}

. To fill the remaining

\frac{1}{2}

of the tank, it will take

\frac{1/2}{1/15} = 7.5\text{ hours}

Adım Adım Çözüm

Determine the combined work rate of all three pumps working together.

Since all three pumps fill

\frac{1}{2}

of the tank in

5\text{ hours}

, their combined rate is

r_1 + r_2 + r_3 = \frac{1/2}{5} = \frac{1}{10}\text{ tank per hour}

Work rate is defined as the fraction of work completed per unit time.

Find the individual work rates of pumps

P_1

and

P_3

r_3 = (r_1 + r_2 + r_3) - (r_1 + r_2) = \frac{1}{10} - \frac{1}{12} = \frac{1}{60}\text{ tank per hour}

. Also,

r_1 = (r_1 + r_2 + r_3) - (r_2 + r_3) = \frac{1}{10} - \frac{1}{20} = \frac{1}{20}\text{ tank per hour}

Subtracting known pair rates from the total three-pump rate yields individual rates.

Calculate the combined work rate of

P_1

and

P_3

r_1 + r_3 = \frac{1}{20} + \frac{1}{60} = \frac{3 + 1}{60} = \frac{4}{60} = \frac{1}{15}\text{ tank per hour}

The rate of two pumps operating together is the sum of their individual rates.

Compute the time needed to fill the remaining portion of the tank.

Remaining fraction to fill is

1 - \frac{1}{2} = \frac{1}{2}

. Time required =

\frac{1/2}{1/15} = \frac{15}{2} = 7.5\text{ hours}

Dividing the remaining workload by the combined rate gives the additional operating time.

Anahtar Kavram

Work-rate problems involving simultaneous rates and partial work completion

Tahmini Süre:2m 0s

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