An agricultural processing facility uses three independent conveyor systems—Conveyor X, Co… — GMAT

An agricultural processing facility uses three independent conveyor systems—Conveyor $X$ , Conveyor $Y$ , and Conveyor $Z$ —to fill a grain storage silo. Working together at their respective constant rates, Conveyors $X$ and $Y$ can fill the empty silo in $12$ hours, while Conveyors $Y$ and $Z$ working together can fill the empty silo in $20$ hours. Initially, Conveyors $X$ and $Z$ work together for $5$ hours, completing exactly $\frac{1}{3}$ of the silo. How many hours would it take Conveyor $Y$ operating alone to fill the remaining $\frac{2}{3}$ of the silo?

Cevap: 20 hours

Cevap

It would take Conveyor

Y

operating alone

20

hours to fill the remaining

\frac{2}{3}

of the silo.

By representing the rate of each pair of conveyors as a fraction of the total job per hour, we find

R_X + R_Y = 1/12

R_Y + R_Z = 1/20

, and

R_X + R_Z = 1/15

. Summing these three equations yields

2(R_X + R_Y + R_Z) = 1/5

, so

R_X + R_Y + R_Z = 1/10

. Subtracting

R_X + R_Z = 1/15

gives

R_Y = 1/30

silo per hour. To fill the remaining

2/3

of the silo, Conveyor

Y

requires

(2/3) / (1/30) = 20

hours.

Adım Adım Çözüm

Express the combined rate of each pair of conveyors as a fraction of the silo filled per hour.

R_X + R_Y = \frac{1}{12}

R_Y + R_Z = \frac{1}{20}

, and

R_X + R_Z = \frac{1/3}{5} = \frac{1}{15}

Work rate equals work done divided by time taken.

Sum the three pairwise rates to determine the combined rate of all three conveyors working together.

2(R_X + R_Y + R_Z) = \frac{1}{12} + \frac{1}{20} + \frac{1}{15} = \frac{12}{60} = \frac{1}{5} \implies R_X + R_Y + R_Z = \frac{1}{10}

Adding pairwise rates counts each conveyor's individual rate twice.

Subtract the combined rate of Conveyors

X

and

Z

from the total rate of all three conveyors to isolate the rate of Conveyor

Y

R_Y = \frac{1}{10} - \frac{1}{15} = \frac{3 - 2}{30} = \frac{1}{30}

of the silo per hour.

Subtracting

(R_X + R_Z)

from

(R_X + R_Y + R_Z)

yields

R_Y

directly.

Divide the remaining fraction of work by Conveyor

Y

's individual rate to find the required time.

\text{Time} = \frac{2/3}{1/30} = 20

hours.

Time required equals remaining work divided by the individual work rate.

Anahtar Kavram

Solving systems of simultaneous work rate equations by summing pairwise rates.

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