A textile manufacturing plant uses two automated weaving looms, Loom P and Loom Q, to prod… — GMAT

A textile manufacturing plant uses two automated weaving looms, Loom $P$ and Loom $Q$ , to produce standardized fabric orders. Working alone at its constant rate, Loom $P$ can complete a full fabric order in $8$ hours, while Loom $Q$ working alone at its constant rate can complete the same order in $12$ hours. Loom $P$ begins working on a full order alone. After $3$ hours, Loom $Q$ joins Loom $P$ , and both looms work together at their respective constant rates until the order is completed. How many total hours does it take, from the moment Loom $P$ begins, to complete the entire fabric order?

A
$3$
$6$ Answer
C
$7.5$
D
$10$
E
$15.5$

Answer

6

hours

In the first

3

hours, Loom

P

completes

3 \times \frac{1}{8} = \frac{3}{8}

of the job, leaving

\frac{5}{8}

of the job remaining. When Loom

Q

joins, their combined rate is

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

of the job per hour. The time needed to complete the remaining

\frac{5}{8}

of the job is

\frac{5/8}{5/24} = 3

hours. Adding the initial

3

hours of solo work gives a total of

6

hours.

Step-by-Step Solution

Determine the individual hourly work rates of Loom

P

and Loom

Q

Rate of Loom

P = \frac{1}{8}

order/hour; Rate of Loom

Q = \frac{1}{12}

order/hour.

Work rate is the reciprocal of the total time needed to complete one full unit of work.

Calculate the fraction of the order completed by Loom

P

during its

3

hours of solo operation.

Work completed by Loom

P = 3 \times \frac{1}{8} = \frac{3}{8}

of the order.

Work done equals rate multiplied by time spent working.

Find the remaining fraction of the order to be completed.

Remaining work =

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

of the order.

The total job represents

1

whole unit of work.

Calculate the combined work rate when both looms operate together.

Combined rate =

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

order/hour.

When entities work together, their individual rates add up.

Determine the time required for both looms together to finish the remaining work.

Time together =

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

hours.

Time equals remaining work divided by the combined work rate.

Calculate the total time from start to finish.

Total time =

3 \text{ hours (solo)} + 3 \text{ hours (combined)} = 6 \text{ hours}

The total elapsed time is the sum of the solo operating time and the combined operating time.

Key Concept

Work Rate and Combined Work

Estimated Time:2m 0s

Question

Answer

Step-by-Step Solution

Key Concept

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