Pipes A and B, operating together at their respective constant rates, can fill an empty st… — GRE General Test

Pipes $A$ and $B$ , operating together at their respective constant rates, can fill an empty storage tank in $6$ hours. Pipes $B$ and $C$ , operating together at their constant rates, can fill the same tank in $8$ hours. The rate at which Pipe $A$ fills the tank is $1.5$ times the rate at which Pipe $C$ fills the tank. If all three pipes operate together for $3$ hours to fill the empty tank, after which Pipe $B$ is closed, how many additional hours will it take for Pipes $A$ and $C$ together to finish filling the tank?

$\frac{6}{5}$ Cevap
B
$\frac{12}{5}$
C
$\frac{18}{5}$
D
$\frac{21}{5}$
E
$3$

Cevap

\frac{6}{5}

hours

The rate equations established from the problem yield individual rates of

\frac{1}{8}

tank/hr for Pipe

A

\frac{1}{24}

tank/hr for Pipe

B

, and

\frac{1}{12}

tank/hr for Pipe

C

. Working together for

3

hours at a combined rate of

\frac{1}{4}

tank/hr, all three pipes fill

\frac{3}{4}

of the tank, leaving

\frac{1}{4}

of the capacity to be filled. After Pipe

B

closes, Pipes

A

and

C

work at a combined rate of

\frac{1}{8} + \frac{1}{12} = \frac{5}{24}

tank/hr. Dividing the remaining

\frac{1}{4}

tank by

\frac{5}{24}

tank/hr gives

\frac{6}{5}

hours.

Adım Adım Çözüm

Express given conditions in terms of work rates per hour (

r_A, r_B, r_C

r_A + r_B = \frac{1}{6}

r_B + r_C = \frac{1}{8}

, and

r_A = 1.5 r_C = \frac{3}{2} r_C

Filling a full tank in

H

hours means completing

\frac{1}{H}

of the tank per hour.

Solve the system of equations for individual rates

r_A, r_B, r_C

Substituting

r_B = \frac{1}{8} - r_C

into

r_A + r_B = \frac{1}{6}

yields

\frac{3}{2} r_C + \frac{1}{8} - r_C = \frac{1}{6} \implies \frac{1}{2} r_C = \frac{1}{24} \implies r_C = \frac{1}{12}

. Consequently,

r_A = \frac{1}{8}

and

r_B = \frac{1}{24}

Finding individual unit rates allows calculating any combined work scenario.

Calculate the fraction of the tank filled in the first 3 hours with all three pipes open.

Combined rate

r_A + r_B + r_C = \frac{1}{8} + \frac{1}{24} + \frac{1}{12} = \frac{3+1+2}{24} = \frac{6}{24} = \frac{1}{4}

of the tank per hour. In

3

hours,

3 \times \frac{1}{4} = \frac{3}{4}

of the tank is filled, leaving

1 - \frac{3}{4} = \frac{1}{4}

of the tank empty.

Work completed equals rate multiplied by time.

Determine the additional time needed for Pipes

A

and

C

to fill the remaining

\frac{1}{4}

of the tank.

Combined rate of

A

and

C

r_A + r_C = \frac{1}{8} + \frac{1}{12} = \frac{5}{24}

per hour. Additional time

t = \frac{\frac{1}{4}}{\frac{5}{24}} = \frac{1}{4} \times \frac{24}{5} = \frac{6}{5}

hours.

Time required equals remaining work divided by active rate.

Anahtar Kavram

Combined Work Rates and Systems of Rate Equations

Soru

Cevap

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