A commercial bakery uses two automated dough mixing machines, Machine A and Machine B, to … — GMAT

A commercial bakery uses two automated dough mixing machines, Machine $A$ and Machine $B$ , to process large batches of dough. Working alone at its constant rate, Machine $A$ can process a full batch of dough in $12$ hours. Machine $B$ , working alone at its constant rate, can process a full batch of dough in $6$ hours. Machine $A$ starts processing a batch alone. After working for $3$ hours, Machine $B$ joins Machine $A$ , and both machines work together at their respective constant rates to finish the batch. What is the total number of hours required to process the entire batch of dough from start to finish?

Answer: 6 hours

Answer

The total number of hours required to process the entire batch of dough from start to finish is 6 hours.

Machine A processes 1/12 of the batch per hour. In the first 3 hours, Machine A finishes 3/12 = 1/4 of the batch, leaving 3/4 of the batch uncompleted. When Machine B joins, the combined processing rate is 1/12 + 1/6 = 1/4 batch per hour. Dividing the remaining 3/4 batch by the combined rate of 1/4 batch per hour yields 3 hours. Adding the initial 3 hours worked by Machine A alone gives a total elapsed time of 6 hours.

Step-by-Step Solution

Determine the individual work rates for Machine

A

and Machine

B

Machine

A

's rate is

\frac{1}{12}

batch/hour, and Machine

B

's rate is

\frac{1}{6}

batch/hour.

Work rate is the fraction of the job completed per hour (1 divided by total hours required alone).

Calculate the portion of the batch completed by Machine

A

operating alone during the first 3 hours.

Machine

A

completes

3 \times \frac{1}{12} = \frac{3}{12} = \frac{1}{4}

of the batch.

Work completed equals work rate multiplied by time worked.

Determine the remaining fraction of the batch to be completed.

Remaining work is

1 - \frac{1}{4} = \frac{3}{4}

of the batch.

The entire job corresponds to 1 whole batch.

Calculate the combined work rate of Machine

A

and Machine

B

Combined rate

= \frac{1}{12} + \frac{1}{6} = \frac{1}{12} + \frac{2}{12} = \frac{3}{12} = \frac{1}{4}

batch per hour.

When machines work together, their individual rates add up.

Calculate the time required for both machines working together to complete the remaining work.

Time together

= \frac{\text{Remaining Work}}{\text{Combined Rate}} = \frac{3/4}{1/4} = 3

hours.

Time equals work divided by rate.

Calculate the total time required from start to finish.

Total time

= 3 \text{ hours (Machine A alone)} + 3 \text{ hours (both together)} = 6

hours.

The total time is the sum of time spent in each phase of the process.

Key Concept

Work Rate and Combined Work

Question

Answer

Step-by-Step Solution

Key Concept

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