Machine X and Machine Y, operating independently at their respective constant rates, can c… — GMAT

Machine X and Machine Y, operating independently at their respective constant rates, can complete a bottling order together in $6$ hours. If Machine X works alone for $2$ hours and is then joined by Machine Y, both machines work together for an additional $4.8$ hours to finish the remaining part of the order. How many hours would it take Machine Y, operating alone at its constant rate, to complete the entire bottling order?

A
$10$
B
$12$
$15$ Answer
D
$16$
E
$18$

Answer

15

hours

The correct answer is

15

hours. Using the work equation

\text{Work} = \text{Rate} \times \text{Time}

, the combined rate of Machine X and Machine Y is

\frac{1}{6}

. The work completed by both machines in the

4.8

-hour period is

4.8 \times \frac{1}{6} = 0.8

of the job. Since the remaining

0.2

of the job was completed by Machine X in

2

hours, Machine X's rate is

\frac{0.2}{2} = 0.1 = \frac{1}{10}

per hour. Subtracting Machine X's rate from the combined rate yields Machine Y's rate:

\frac{1}{6} - \frac{1}{10} = \frac{1}{15}

per hour. Therefore, Machine Y takes

15

hours operating alone to finish the entire bottling order.

Step-by-Step Solution

Define rates for Machine X and Machine Y

Let Machine X's rate be

r_x

orders per hour and Machine Y's rate be

r_y

orders per hour. Their combined rate is

r_x + r_y = \frac{1}{6}

orders per hour.

Combined work rate is the reciprocal of the combined completion time of

6

hours.

Express the total work completed in two stages

Machine X works alone for

2

hours completing

2 r_x

of the order. Then both work together for

4.8

hours completing

4.8(r_x + r_y)

of the order. Thus,

2 r_x + 4.8(r_x + r_y) = 1

The sum of work completed in the two stages equals

1

full job.

Substitute the combined rate into the equation to find

r_x

Since

r_x + r_y = \frac{1}{6}

, we substitute:

2 r_x + 4.8\left(\frac{1}{6}\right) = 1 \implies 2 r_x + 0.8 = 1 \implies 2 r_x = 0.2 \implies r_x = 0.1 = \frac{1}{10}

Substituting the combined rate simplifies the equation to a single variable,

r_x

Calculate

r_y

and the time needed for Machine Y working alone

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

orders per hour. Time taken by Machine Y alone

= \frac{1}{r_y} = 15

hours.

Subtracting Machine X's rate from the combined rate yields Machine Y's rate, whose reciprocal gives the time to complete the job alone.

Key Concept

Work Rate and Combined Work

Estimated Time:2m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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