Two automated document redaction pipelines, Pipeline A and Pipeline B, process digital leg… — GMAT

Two automated document redaction pipelines, Pipeline $A$ and Pipeline $B$ , process digital legal archives. Working alone at its constant rate, Pipeline $A$ can redact a standard archive in $8$ hours. Working alone at its constant rate, Pipeline $B$ can redact the same standard archive in $12$ hours. Both pipelines begin processing a standard archive together. After $3$ hours of joint operation, Pipeline $A$ encounters a system error and stops. Pipeline $B$ continues working alone at its constant rate until the entire archive is redacted. How many total hours does it take from start to finish to complete the redaction of the archive?

Cevap: 7.5 hours

Cevap

The total time required to redact the entire archive from start to finish is 7.5 hours.

To find the total time required, first determine the combined rate of Pipeline A and Pipeline B:

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

archive per hour. In the first 3 hours of joint work, they complete

3 \times \frac{5}{24} = \frac{5}{8}

of the job, leaving

1 - \frac{5}{8} = \frac{3}{8}

of the archive remaining. Pipeline B processes the remaining

\frac{3}{8}

at its individual rate of

\frac{1}{12}

per hour, taking

\frac{3/8}{1/12} = 4.5

hours. Adding the initial 3 hours to the 4.5 additional hours yields a total of 7.5 hours.

Adım Adım Çözüm

Determine the individual work rate for each pipeline.

Pipeline A's rate is

\frac{1}{8}

archive per hour; Pipeline B's rate is

\frac{1}{12}

archive per hour.

Work rate is the fraction of job completed per hour (

1 / \text{time required alone}

Calculate the combined work rate during joint operation.

Combined rate is

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

archive per hour.

When entities work concurrently, their rates are added together.

Determine the amount of work completed in the first 3 hours.

Work completed is

3 \times \frac{5}{24} = \frac{5}{8}

of the total archive.

Work done equals rate multiplied by time.

Find the fraction of the archive remaining to be redacted.

Remaining work is

1 - \frac{5}{8} = \frac{3}{8}

of the archive.

The full task represents 1 complete unit of work.

Calculate how long Pipeline B needs to finish the remaining work alone.

Solo time for Pipeline B is

\frac{3/8}{1/12} = 4.5

hours.

Time equals remaining work divided by Pipeline B's individual rate.

Sum the joint operation time and the solo operation time.

Total time is

3 + 4.5 = 7.5

hours.

The overall duration consists of two sequential periods: 3 hours working together plus 4.5 hours for Pipeline B working alone.

Anahtar Kavram

Combined Work Rates and Multi-Stage Work Scenarios

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