Scanner Unit Alpha and Scanner Unit Beta, operating independently at their respective cons… — GMAT

Scanner Unit Alpha and Scanner Unit Beta, operating independently at their respective constant rates, can complete a document digitization project together in $12$ hours. Scanner Unit Alpha operates at a rate that is $50\%$ faster than that of Scanner Unit Beta. The project is carried out in three consecutive stages: first, Scanner Unit Alpha operates alone for $4$ hours; second, Scanner Unit Beta joins Alpha and both operate together until $70\%$ of the entire project is completed; finally, Scanner Unit Beta finishes the remaining portion of the project alone. How many total hours does it take to complete the entire digitization project?

Cevap: 19 hours

Cevap

The total time required to complete the entire digitization project is 19 hours.

To find the total time needed, calculate the individual rates first. With a combined rate of

r_A + r_B = \frac{1}{12}

and

r_A = 1.5 r_B

, solving

2.5 r_B = \frac{1}{12}

yields

r_B = \frac{1}{30}

and

r_A = \frac{1}{20}

. During Stage 1 (

4

hours), Alpha completes

4 \times \frac{1}{20} = 0.20

of the job. In Stage 2, both units work together to bring completion from

20\%

70\%

(

0.50

work), taking

\frac{0.50}{1/12} = 6

hours. In Stage 3, Beta completes the remaining

0.30

work alone, taking

\frac{0.30}{1/30} = 9

hours. Summing all stage durations yields

4 + 6 + 9 = 19

hours.

Adım Adım Çözüm

Determine the individual work rates of Scanner Unit Alpha (

r_A

) and Scanner Unit Beta (

r_B

r_B = \frac{1}{30}

project per hour, and

r_A = \frac{1}{20}

project per hour.

Since their combined rate is

\frac{1}{12}

project per hour and

r_A = 1.5 r_B

, we solve

2.5 r_B = \frac{1}{12}

to find

r_B = \frac{1}{30}

and

r_A = \frac{1}{20}

Calculate the fraction of work completed during Stage 1.

Scanner Unit Alpha completes

0.20

(

20\%

) of the project in

4

hours.

Alpha works alone for

4

hours at a rate of

\frac{1}{20}

project per hour:

4 \times \frac{1}{20} = 0.20

Calculate the duration of Stage 2 where both units work together.

Stage 2 takes

6

hours.

The combined units must complete the portion from

20\%

70\%

, which represents

0.70 - 0.20 = 0.50

of the project. At a combined rate of

\frac{1}{12}

project per hour, the time required is

\frac{0.50}{1/12} = 6

hours.

Calculate the duration of Stage 3 where Scanner Unit Beta works alone.

Stage 3 takes

9

hours.

Beta must complete the remaining

30\%

(

0.30

) of the project alone. At a rate of

\frac{1}{30}

project per hour, the time required is

\frac{0.30}{1/30} = 9

hours.

Sum the durations of all three stages to determine total project time.

Total time =

19

hours.

Adding the duration of each stage:

4 \text{ hours} + 6 \text{ hours} + 9 \text{ hours} = 19 \text{ hours}

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Work Rate and Combined Work

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