Working alone at their respective constant rates, Machine A can complete a production orde… — GRE General Test

Working alone at their respective constant rates, Machine $A$ can complete a production order in $6\text{ hours}$ , Machine $B$ in $8\text{ hours}$ , and Machine $C$ in $12\text{ hours}$ . All three machines start working together on the order at 9:00 AM. At 10:00 AM, Machine $A$ breaks down and stops working, while Machines $B$ and $C$ continue working together at their constant rates until the order is completed. At what time will the production order be completed?

A
11:40 AM
B
12:00 PM
1:00 PMAnswer
D
1:48 PM
E
2:12 PM

Answer

The production order will be completed at 1:00 PM.

The option specifying 1:00 PM is correct because during the first hour (9:00 AM to 10:00 AM), all three machines work together and complete

\frac{1}{6} + \frac{1}{8} + \frac{1}{12} = \frac{3}{8}

of the job. This leaves

\frac{5}{8}

of the job remaining at 10:00 AM. Machines B and C work together at a rate of

\frac{1}{8} + \frac{1}{12} = \frac{5}{24}

of the job per hour. Dividing the remaining work (

\frac{5}{8}

) by this combined rate (

\frac{5}{24}

) gives exactly 3 hours. Adding 3 hours to 10:00 AM yields 1:00 PM.

Step-by-Step Solution

Calculate individual work rates for each machine.

Machine A rate =

\frac{1}{6}

order/hr, Machine B rate =

\frac{1}{8}

order/hr, Machine C rate =

\frac{1}{12}

order/hr.

Work rate is defined as job completed per unit of time.

Calculate the work completed by all three machines from 9:00 AM to 10:00 AM (1 hour).

Combined rate =

\frac{1}{6} + \frac{1}{8} + \frac{1}{12} = \frac{4 + 3 + 2}{24} = \frac{9}{24} = \frac{3}{8}

of the order.

All three machines work together for exactly 1 hour.

Determine the remaining fraction of the order after 10:00 AM.

Remaining work =

1 - \frac{3}{8} = \frac{5}{8}

of the order.

Subtract completed work from the total work (1 whole order).

Calculate the combined rate of Machines B and C.

Combined rate of B and C =

\frac{1}{8} + \frac{1}{12} = \frac{3 + 2}{24} = \frac{5}{24}

order/hr.

Machine A stops, leaving only B and C working.

Calculate the additional time needed to complete the remaining work.

Time =

\frac{5/8}{5/24} = \frac{5}{8} \times \frac{24}{5} = 3\text{ hours}

Divide remaining work by the combined rate of the remaining active machines.

Add the additional time to the breakdown time (10:00 AM).

Completion time = 10:00 AM + 3 hours = 1:00 PM.

The remaining work begins at 10:00 AM when Machine A breaks down.

Key Concept

Combined Work Rates and Staggered Work Times

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