A software engineering team uses three automated refactoring scripts—Script A, Script B, a… — GMAT

A software engineering team uses three automated refactoring scripts—Script A, Script B, and Script C—to update a large codebase. Working alone at their respective constant rates, Script A can finish the entire codebase in $10$ hours, Script B in $15$ hours, and Script C in $30$ hours. All three scripts begin running simultaneously. After $2$ hours, Script A encounters an error and stops running. Script B and Script C continue running together without interruption until the codebase is completely refactored. How many additional hours will it take for Script B and Script C to complete the remaining work?

Cevap: 6 hours

Cevap

6 hours

The rate of Script A is

\frac{1}{10}

of the codebase per hour, Script B is

\frac{1}{15}

per hour, and Script C is

\frac{1}{30}

per hour. Working together for

2

hours, all three scripts complete

2 \times \left(\frac{1}{10} + \frac{1}{15} + \frac{1}{30}\right) = 2 \times \frac{1}{5} = \frac{2}{5}

of the codebase, leaving

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

of the task remaining. With Script A offline, the combined rate of Script B and Script C is

\frac{1}{15} + \frac{1}{30} = \frac{1}{10}

per hour. The additional time required to complete the remaining work is

\frac{3/5}{1/10} = 6

hours.

Adım Adım Çözüm

Determine the individual work rates of Script A, Script B, and Script C.

Rate of Script A =

\frac{1}{10}

codebase/hour, Rate of Script B =

\frac{1}{15}

codebase/hour, Rate of Script C =

\frac{1}{30}

codebase/hour.

Work rate is the reciprocal of the total time required to complete one entire job working alone.

Calculate the combined work rate of all three scripts during the initial 2-hour period.

Combined Rate(A+B+C) =

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

codebase/hour.

When multiple agents work together simultaneously, their individual work rates are added.

Determine the fraction of the codebase refactored during the first 2 hours.

Work Completed =

2 \text{ hours} \times \frac{1}{5} \text{ codebase/hour} = \frac{2}{5}

of the total codebase.

Total work done equals rate multiplied by time.

Calculate the remaining fraction of the codebase left to be processed.

Remaining Work =

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

of the total codebase.

Subtracting the completed fraction from 1 gives the uncompleted portion of the job.

Calculate the combined rate of Script B and Script C after Script A stops.

Rate(B+C) =

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

codebase/hour.

Only Script B and Script C contribute to the remaining work.

Calculate the additional time required for Script B and Script C to finish the remaining work.

Additional Time =

\frac{3/5}{1/10} = \frac{3}{5} \times 10 = 6

hours.

Time equals remaining work divided by the active combined rate.

Anahtar Kavram

Work Rate and Combined Work

Tahmini Süre:1m 30s

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Cevap

Adım Adım Çözüm

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