Scanner X can scan a batch of archival documents in 6 hours operating alone at a constant … — GMAT

Scanner X can scan a batch of archival documents in $6$ hours operating alone at a constant rate. Scanner Y can scan the exact same batch of documents in $3$ hours operating alone at a constant rate. Working together simultaneously at their respective constant rates, how many hours will it take Scanner X and Scanner Y to scan the entire batch of documents?

Answer: 2 hours

Answer

It will take Scanner X and Scanner Y a total of

2

hours to scan the batch of documents together.

To calculate combined completion time, add the rates of each machine rather than their completion times. Scanner X completes

\frac{1}{6}

of the task per hour, and Scanner Y completes

\frac{1}{3}

(or

\frac{2}{6}

) of the task per hour. Together, they complete

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

of the task per hour. The inverse of this rate gives the total time required:

2

hours.

Step-by-Step Solution

Calculate individual work rates per hour.

Scanner X rate =

\frac{1}{6}

batch/hour, Scanner Y rate =

\frac{1}{3}

batch/hour.

Work rate is defined as

\text{Rate} = \frac{\text{Work}}{\text{Time}}

Add the individual rates together to find the total combined rate.

Combined rate =

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

batch/hour.

When entities work together simultaneously, their work rates are additive.

Compute total time taken for 1 whole batch.

Total Time =

2

hours.

Time is the reciprocal of the combined work rate:

\text{Time} = \frac{1}{\text{Combined Rate}}

Key Concept

Combined Work Rate

Estimated Time:45s

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