Working alone at its constant rate, Printer X can complete a printing job in 4\text{ hours… — GRE General Test

Working alone at its constant rate, Printer $X$ can complete a printing job in $4\text{ hours}$ . Working alone at its constant rate, Printer $Y$ can complete the same printing job in $6\text{ hours}$ . Printer $X$ begins working on the job alone and works for $1\text{ hour}$ . At that point, Printer $Y$ joins Printer $X$ , and both printers work together at their respective constant rates until the job is completed. What is the total time, in hours, required to complete the entire job from start to finish?

A
$1.8\text{ hours}$
B
$2.4\text{ hours}$
$2.8\text{ hours}$ Cevap
D
$3.2\text{ hours}$
E
$3.5\text{ hours}$

Cevap

2.8\text{ hours}

The correct answer is

2.8\text{ hours}

. Printer

X

works alone for

1\text{ hour}

at a rate of

\frac{1}{4}

job per hour, completing

\frac{1}{4}

of the total job. This leaves

\frac{3}{4}

of the job unfinished. When Printer

Y

joins, their combined rate is

\frac{1}{4} + \frac{1}{6} = \frac{5}{12}

job per hour. Dividing the remaining

\frac{3}{4}

of the job by

\frac{5}{12}

yields

\frac{3}{4} \times \frac{12}{5} = \frac{9}{5} = 1.8\text{ hours}

for the joint work phase. Adding the initial

1\text{ hour}

of solo work yields a total of

1 + 1.8 = 2.8\text{ hours}

Adım Adım Çözüm

Calculate individual work rates and the portion of the job completed in the first hour

Printer

X

's rate is

\frac{1}{4}

job/hour and Printer

Y

's rate is

\frac{1}{6}

job/hour. In the first hour, Printer

X

completes

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

of the job.

Printer

X

works alone for the first hour before Printer

Y

joins.

Determine the remaining fraction of the job

Remaining job =

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

The entire job is represented by

1

, so subtracting the completed portion yields the remaining portion.

Calculate the combined rate of both printers working together

Combined rate =

\frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12}

job/hour.

Rates add when workers or machines work simultaneously.

Calculate the time required for both printers to finish the remaining job and find total time

Time together =

\frac{3/4}{5/12} = \frac{3}{4} \times \frac{12}{5} = \frac{9}{5} = 1.8\text{ hours}

. Total time =

1 + 1.8 = 2.8\text{ hours}

Time equals remaining work divided by combined rate, plus the

1\text{ hour}

already elapsed.

Anahtar Kavram

Combined Work Rates and Multi-Stage Work Problems

Alternatif Yöntem

Convert the job into arbitrary work units. Let the job equal

12\text{ units}

(the LCM of

4

and

6

). Printer

X

produces

12 / 4 = 3\text{ units/hour}

and Printer

Y

produces

12 / 6 = 2\text{ units/hour}

. In the first hour, Printer

X

produces

3\text{ units}

, leaving

12 - 3 = 9\text{ units}

. Working together, their combined rate is

3 + 2 = 5\text{ units/hour}

. The remaining

9\text{ units}

take

9 / 5 = 1.8\text{ hours}

. Total time is

1 + 1.8 = 2.8\text{ hours}

Tahmini Süre:1m 30s

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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Alternatif Yöntem

Konular

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