An archival digitization laboratory uses two high-resolution scanner units, Unit A and Uni… — GMAT

An archival digitization laboratory uses two high-resolution scanner units, Unit $A$ and Unit $B$ , to process historic document collections. Working alone at its constant rate, Unit $A$ can digitize a standard collection in $24$ hours. Working alone at its constant rate, Unit $B$ can digitize the same collection in $40$ hours. Unit $A$ begins digitizing a standard collection alone. After $8$ hours of operation, Unit $B$ is turned on, and both units work together at their respective constant rates until the entire collection is digitized. How many total hours does it take to digitize the collection from start to finish?

Answer: 18 hours

Answer

The total time required to digitize the collection from start to finish is 18 hours.

Unit A completes

\frac{1}{3}

of the total job during its 8 solo hours. The remaining

\frac{2}{3}

of the job is completed by Units A and B working together at a combined rate of

\frac{1}{15}

per hour, taking 10 hours. Adding the 8 solo hours to the 10 combined hours yields a total time of 18 hours.

Step-by-Step Solution

Determine the individual work rates per hour for each unit.

Unit A's rate is

\frac{1}{24}

of the collection per hour, and Unit B's rate is

\frac{1}{40}

of the collection per hour.

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

R = \frac{1}{T}

Calculate the fraction of the work completed by Unit A alone during the first 8 hours.

Work completed

= 8 \times \frac{1}{24} = \frac{8}{24} = \frac{1}{3}

of the collection.

Work done equals rate multiplied by time (

W = R \times t

Calculate the remaining fraction of the collection to be digitized.

Remaining work

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

of the collection.

The total job represents 1 whole unit.

Calculate the combined work rate when both units operate together.

Combined rate

= \frac{1}{24} + \frac{1}{40} = \frac{5}{120} + \frac{3}{120} = \frac{8}{120} = \frac{1}{15}

of the collection per hour.

When working simultaneously, individual rates add together.

Find the time required for both units working together to complete the remaining work.

Combined time

= \frac{2/3}{1/15} = \frac{2}{3} \times 15 = 10

hours.

Time equals remaining work divided by the combined rate (

t = \frac{W_{\text{remaining}}}{R_{\text{combined}}}

Calculate the total time required from start to finish.

Total time

= 8 \text{ hours} + 10 \text{ hours} = 18

hours.

Total time is the sum of the time Unit A worked alone and the time both units worked together.

Key Concept

Work Rate and Combined Work

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