An agricultural facility uses two automated loading conveyors, Conveyor A and Conveyor B, … — GMAT

An agricultural facility uses two automated loading conveyors, Conveyor $A$ and Conveyor $B$ , to fill a grain storage silo. Working alone at its constant rate, Conveyor $A$ can fill the silo in 10 hours. Working alone at its constant rate, Conveyor $B$ can fill the silo in 15 hours. Conveyor $A$ begins filling the empty silo alone. After 4 hours, Conveyor $B$ is turned on, and both conveyors work together at their respective constant rates until the silo is completely full. How many total hours does it take, from the moment Conveyor $A$ starts, to completely fill the silo?

Answer: 7.6 hours

Answer

The total time required to fill the silo is 7.6 hours.

Conveyor A fills 1/10 of the silo per hour and works alone for 4 hours, completing 2/5 of the total job. The remaining 3/5 of the job is completed by both conveyors working together at a combined rate of 1/10 + 1/15 = 1/6 silo per hour. Dividing 3/5 by 1/6 gives 3.6 hours for the second stage. Adding the initial 4 hours yields a total time of 7.6 hours.

Step-by-Step Solution

Determine the individual hourly rates of Conveyor A and Conveyor B.

Conveyor A rate =

1/10

silo per hour; Conveyor B rate =

1/15

silo per hour.

Work rate is the reciprocal of the time required to complete the total job.

Calculate the fraction of the silo filled by Conveyor A working alone for 4 hours.

Work completed =

4 \times \frac{1}{10} = \frac{2}{5}

of the silo.

Work done equals rate multiplied by time.

Determine the remaining fraction of work to be completed.

Remaining work =

1 - \frac{2}{5} = \frac{3}{5}

of the silo.

Subtract the completed fraction from the total job (1).

Calculate the combined work rate when both conveyors operate together.

Combined rate =

\frac{1}{10} + \frac{1}{15} = \frac{3 + 2}{30} = \frac{5}{30} = \frac{1}{6}

silo per hour.

When working together, individual rates add up.

Determine the time required for both conveyors to finish the remaining work.

Time =

\frac{3/5}{1/6} = \frac{18}{5} = 3.6

hours.

Divide remaining work by the combined work rate.

Sum the time spent in both stages to find the total time.

Total time =

4 + 3.6 = 7.6

hours.

The total duration includes 4 hours of Conveyor A operating alone plus 3.6 hours of combined operation.

Key Concept

Work Rate and Combined Work

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