In a manufacturing facility, three automated machines—Machine A, Machine B, and Machine C—… — GRE General Test

In a manufacturing facility, three automated machines—Machine A, Machine B, and Machine C—produce components at constant individual rates. Machine A working alone completes $\frac{3}{10}$ of a standard daily order in $3$ hours. Machine B working alone completes $\frac{2}{5}$ of the order in $6$ hours. Machine C working alone completes $\frac{1}{4}$ of the order in $5$ hours.

Initially, all three machines work together for $3$ hours. At the end of $3$ hours, Machine A malfunctions and stops, while Machines B and C continue working together without interruption until the daily order is finished. How many total hours does it take from the start to complete the entire daily order?

Answer: 6 hours

Answer

The total time required from the start to complete the entire daily order is 6 hours.

Each machine's rate per hour is found by dividing the given fraction by hours: Machine A is

\frac{1}{10}

, B is

\frac{1}{15}

, and C is

\frac{1}{20}

. Summing these gives an initial combined rate of

\frac{13}{60}

per hour. In the first 3 hours, the three machines complete

3 \times \frac{13}{60} = \frac{13}{20}

of the order, leaving

1 - \frac{13}{20} = \frac{7}{20}

of the order remaining. When Machine A stops, the remaining combined rate of B and C is

\frac{1}{15} + \frac{1}{20} = \frac{7}{60}

per hour. Dividing the remaining

\frac{7}{20}

\frac{7}{60}

yields 3 additional hours. Adding the initial 3 hours gives a total duration of 6 hours.

Step-by-Step Solution

Determine individual hourly work rates for each machine.

Machine A rate =

\frac{1}{10}

order/hr, Machine B rate =

\frac{1}{15}

order/hr, Machine C rate =

\frac{1}{20}

order/hr.

Divide the fraction of the job completed by the duration in hours to find the unit rate for each machine.

Calculate the combined rate of all three machines and the fraction completed in the first 3 hours.

Combined rate =

\frac{13}{60}

order/hr; Work completed in 3 hours =

\frac{13}{20}

, leaving

\frac{7}{20}

of the order unfinished.

Sum the three rates using a common denominator of 60, then multiply by 3 hours to find the total work done during the first stage.

Calculate the combined rate of Machines B and C, and find the additional time required to complete the remaining fraction.

Combined rate of B and C =

\frac{7}{60}

order/hr; Additional time needed = 3 hours.

Divide the remaining fraction

\frac{7}{20}

by the combined rate

\frac{7}{60}

to determine the remaining hours needed.

Add the initial duration and additional duration to obtain the total time.

Total time =

3 + 3 = 6

hours.

The question asks for the total elapsed time from the start of the job.

Key Concept

Adding rational fractions with different denominators to solve multi-stage work and rate problems.

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