A specialized express freight facility uses three automated sorting systems—System 1, Syst… — GMAT

A specialized express freight facility uses three automated sorting systems—System 1, System 2, and System 3—to process incoming shipments. Operating continuously at their respective constant rates, System 1 and System 2 working together can process 1 full shipment in 6 hours; System 2 and System 3 working together can process the same shipment in 10 hours; and System 1 and System 3 working together can process the shipment in 7.5 hours. Processing begins with System 1 and System 2 working together. After 3 hours, System 1 breaks down and stops operating, at which point System 3 is immediately activated to work alongside System 2 until the entire shipment is completed. How many total hours does it take, from the start of processing, to complete the entire shipment?

Cevap: 8 hours

Cevap

The total time required from start to completion is 8 hours.

Converting completion times to work rates per hour gives paired rates of 1/6, 1/10, and 2/15. Summing these and dividing by 2 yields a combined three-system rate of 1/5 shipment per hour. In the first 3 hours, System 1 and System 2 complete 3 * (1/6) = 1/2 of the shipment. System 2 and System 3 then finish the remaining 1/2 at their combined rate of 1/10 per hour, requiring (1/2) / (1/10) = 5 hours. Total elapsed time is 3 + 5 = 8 hours.

Adım Adım Çözüm

Set up equations for the rate of work done per hour by each pair of systems

Let

r_1, r_2, r_3

be the individual work rates in shipments per hour. Then

r_1 + r_2 = \frac{1}{6}

r_2 + r_3 = \frac{1}{10}

, and

r_1 + r_3 = \frac{1}{7.5} = \frac{2}{15}

Work rate is inversely proportional to completion time (

Rate = \frac{Work}{Time}

Calculate the combined processing rate of all three systems

2(r_1 + r_2 + r_3) = \frac{1}{6} + \frac{1}{10} + \frac{2}{15} = \frac{5 + 3 + 4}{30} = \frac{12}{30} = \frac{2}{5}

, which simplifies to

r_1 + r_2 + r_3 = \frac{1}{5}

shipment per hour.

Adding the three paired rates accounts for each individual system's rate exactly twice.

Determine the amount of work completed during the initial 3-hour period

Work completed =

3 \text{ hours} \times (r_1 + r_2) = 3 \times \frac{1}{6} = \frac{1}{2}

of the total shipment.

System 1 and System 2 operate together at a combined rate of

\frac{1}{6}

shipment per hour for 3 hours.

Calculate the time needed for System 2 and System 3 to complete the remaining shipment

Remaining work =

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

. Time required =

\frac{1/2}{r_2 + r_3} = \frac{1/2}{1/10} = 5

hours.

System 2 and System 3 work together at a combined rate of

\frac{1}{10}

shipment per hour to finish the remaining half of the job.

Calculate the total time elapsed from start to finish

Total time =

3 \text{ hours} + 5 \text{ hours} = 8 \text{ hours}

The question asks for the total duration of the process from the beginning.

Anahtar Kavram

Solving systems of simultaneous work rate equations and analyzing multi-stage combined work.

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