An agricultural facility uses three conveyor belts—X, Y, and Z—to transfer harvested grain… — GRE General Test

An agricultural facility uses three conveyor belts— $X$ , $Y$ , and $Z$ —to transfer harvested grain into a storage silo. Working together at their respective constant rates, Belt $X$ and Belt $Y$ can fill the empty silo in $6\text{ hours}$ . Working together at their respective constant rates, Belt $Y$ and Belt $Z$ can fill the empty silo in $8\text{ hours}$ . If Belt $X$ operates at twice the rate of Belt $Z$ , how many hours would it take Belt $Y$ working alone at its constant rate to fill the empty silo?

A
$10\text{ hours}$
$12\text{ hours}$ Answer
C
$16\text{ hours}$
D
$18\text{ hours}$
E
$24\text{ hours}$

Answer

12\text{ hours}

The correct answer is

12\text{ hours}

. Subtracting the combined rate equation for Belts

Y

and

Z

(

r_Y + r_Z = 1/8

) from the equation for Belts

X

and

Y

(

r_X + r_Y = 1/6

) yields

r_X - r_Z = 1/24

. Since Belt

X

works at twice the rate of Belt

Z

(

r_X = 2r_Z

), substituting gives

r_Z = 1/24

. Substituting

r_Z

back into

r_Y + r_Z = 1/8

gives

r_Y = 1/8 - 1/24 = 1/12

. Therefore, Belt

Y

operating alone takes

12\text{ hours}

to fill the silo.

Step-by-Step Solution

Define the work rates of each conveyor belt.

Let

r_X

r_Y

, and

r_Z

be the fraction of the silo filled per hour by Belts

X

Y

, and

Z

, respectively.

Establishing rates per unit of time allows linear combination of work performed.

Set up equations based on the given combined times and rate relationships.

r_X + r_Y = \frac{1}{6}

r_Y + r_Z = \frac{1}{8}

, and

r_X = 2r_Z

Combined rates equal the reciprocal of the total time required for combined work.

Subtract the second equation from the first to isolate

r_X - r_Z

(r_X + r_Y) - (r_Y + r_Z) = \frac{1}{6} - \frac{1}{8} \implies r_X - r_Z = \frac{4 - 3}{24} = \frac{1}{24}

Eliminating

r_Y

gives a direct linear relationship between

r_X

and

r_Z

Substitute

r_X = 2r_Z

into

r_X - r_Z = \frac{1}{24}

to solve for

r_Z

2r_Z - r_Z = \frac{1}{24} \implies r_Z = \frac{1}{24}

Determines the individual rate of Belt

Z

Substitute

r_Z = \frac{1}{24}

back into the equation

r_Y + r_Z = \frac{1}{8}

to find

r_Y

r_Y = \frac{1}{8} - \frac{1}{24} = \frac{3}{24} - \frac{1}{24} = \frac{2}{24} = \frac{1}{12}

Finds the individual rate of Belt

Y

Calculate the time required for Belt

Y

alone.

\text{Time} = \frac{1}{r_Y} = 12\text{ hours}.

The time to complete one full job is the reciprocal of the individual rate.

Key Concept

Work Rates and Combined Rate Equations

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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