A pharmaceutical production facility utilizes three automated synthesis modules—Module A, … — GMAT

A pharmaceutical production facility utilizes three automated synthesis modules—Module A, Module B, and Module C—to produce a specific batch of medication. Working alone at its constant rate, Module A can complete a full batch in $15$ hours. Modules B and C, working together at their respective constant rates, can complete a full batch in $20$ hours. A production run begins with Module A and Module B working together for $4$ hours. At that point, Module A is turned off for recalibration, and Module C immediately joins Module B to complete the remaining portion of the batch. If Modules B and C work together for $12$ hours to finish the batch, how many hours would it take Module B to complete a full batch working alone?

A
$20$ hours
B
$25$ hours
$30$ hoursCevap
D
$40$ hours
E
$60$ hours

Cevap

Module B would take

30

hours to complete a full batch working alone.

The correct answer is

30

hours. In Stage 2, Modules B and C work together for

12

hours at their combined rate of

\frac{1}{20}

batch per hour, completing

\frac{12}{20} = \frac{3}{5}

of the entire batch. This leaves

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

of the batch that was completed during Stage 1 by Modules A and B working together for

4

hours. Setting up the Stage 1 equation:

4(r_A + r_B) = \frac{2}{5}

. Substituting Module A's rate

r_A = \frac{1}{15}

, we get

4\left(\frac{1}{15} + r_B\right) = \frac{2}{5} \implies \frac{4}{15} + 4r_B = \frac{6}{15} \implies 4r_B = \frac{2}{15} \implies r_B = \frac{1}{30}

. Therefore, Module B takes

30

hours operating alone.

Adım Adım Çözüm

Express the individual and combined rates of the modules

Rate of Module A (

r_A

) =

\frac{1}{15}

batch/hr; Combined rate of Modules B and C (

r_B + r_C

) =

\frac{1}{20}

batch/hr.

Work rate is defined as the fraction of the job completed per unit of time.

Calculate the work completed during Stage 2

Work in Stage 2 =

12 \times (r_B + r_C) = 12 \times \frac{1}{20} = \frac{3}{5}

of the batch.

Modules B and C worked together for

12

hours at their known combined rate of

\frac{1}{20}

batch/hr.

Calculate the work done in Stage 1 and solve for Module B's rate (

r_B

)

Total Work = Stage 1 Work + Stage 2 Work =

1

. Thus,

4(r_A + r_B) + \frac{3}{5} = 1 \implies 4\left(\frac{1}{15} + r_B\right) = \frac{2}{5} \implies \frac{4}{15} + 4r_B = \frac{6}{15} \implies 4r_B = \frac{2}{15} \implies r_B = \frac{1}{30}

batch/hr.

The sum of the work performed across both stages must equal

1

full batch.

Convert Module B's rate into total time required working alone

Time for Module B alone =

\frac{1}{r_B} = 30

hours.

Total time working alone is the reciprocal of the individual work rate.

Anahtar Kavram

Work Rate and Combined Work

Soru

Cevap

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