A pharmaceutical facility uses two automated purification columns, Column X and Column Y, … — GMAT

A pharmaceutical facility uses two automated purification columns, Column $X$ and Column $Y$ , to process standard batches of a synthetic compound. Working alone at its constant rate, Column $X$ can purify a full batch in $15$ hours, while Column $Y$ can purify a full batch in $10$ hours working alone at its constant rate. Column $X$ begins purifying a batch alone. After $3$ hours, Column $Y$ is turned on to work alongside Column $X$ , and both columns work together at their respective constant rates until the batch is completely purified. What is the total time, in hours, from the moment Column $X$ started until the batch is completely purified?

A
4.8 hours
B
7.2 hours
7.8 hoursCevap
D
9.6 hours
E
11.0 hours

Cevap

7.8 hours

The correct answer of 7.8 hours is obtained by evaluating the work completed in stages. In the first 3 hours, Column X completes 3/15 or 1/5 of the batch, leaving 4/5 of the batch remaining. When Column Y joins Column X, their combined rate is 1/15 + 1/10 = 1/6 batch per hour. Completing the remaining 4/5 of the batch at a rate of 1/6 per hour requires (4/5) / (1/6) = 4.8 hours. Adding the initial 3 hours yields a total time of 7.8 hours.

Adım Adım Çözüm

Determine the individual hourly work rates for each column

Rate of Column

X = \frac{1}{15}

batch per hour; Rate of Column

Y = \frac{1}{10}

batch per hour.

Work rate is the reciprocal of total time required to complete one full job.

Calculate the fraction of work completed during the initial stage

Work done by

X = 3 \times \frac{1}{15} = \frac{3}{15} = \frac{1}{5}

of the batch.

Column

X

works alone for the first 3 hours.

Find the remaining fraction of work to be completed

Remaining work

= 1 - \frac{1}{5} = \frac{4}{5}

of the batch.

Subtract the completed fraction from the total job (1 full batch).

Determine the combined work rate of both columns working together

Combined rate

= \frac{1}{15} + \frac{1}{10} = \frac{2}{30} + \frac{3}{30} = \frac{5}{30} = \frac{1}{6}

batch per hour.

Rates add when two entities work simultaneously.

Calculate the time required for both columns to finish the remaining work

Time together

= \frac{\text{Remaining Work}}{\text{Combined Rate}} = \frac{4/5}{1/6} = \frac{4}{5} \times 6 = \frac{24}{5} = 4.8

hours.

Time equals work divided by rate.

Calculate the total elapsed time from the start

Total time

= 3 \text{ hours (alone)} + 4.8 \text{ hours (together)} = 7.8

hours.

Add the initial duration to the combined working duration.

Anahtar Kavram

Work Rate and Combined Work

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru