Working together at their respective constant rates, Machine X and Machine Y can complete … — GMAT

Working together at their respective constant rates, Machine X and Machine Y can complete a specialized production order in $12$ hours. If Machine X works alone for $4$ hours and Machine Y works alone for $9$ hours, they complete $\frac{7}{12}$ of the total order. Machine Z operates at a constant rate that is $50\%$ greater than the rate of Machine X. How many hours would it take Machine Y and Machine Z working together to complete the entire order?

$10$ Answer
B
$12$
C
$15$
D
$20$
E
$40$

Answer

10

hours

The correct answer is

10

hours. Solving the system of equations yields individual rates of

\frac{1}{30}

order/hour for Machine X and

\frac{1}{20}

order/hour for Machine Y. A

50\%

rate increase gives Machine Z a rate of

\frac{1}{20}

order/hour. Adding the rates of Machine Y and Machine Z yields a combined rate of

\frac{1}{10}

order/hour, which requires

10

hours to complete one job.

Step-by-Step Solution

Define variables and model the joint rate equation.

Let

x

be the rate of Machine X (order/hour) and

y

be the rate of Machine Y (order/hour). Their combined rate equation is

x + y = \frac{1}{12}

Working together for

12

hours completes

1

full order.

Set up the second equation based on the partial work completed.

4x + 9y = \frac{7}{12}

Machine X works for

4

hours and Machine Y works for

9

hours to complete

\frac{7}{12}

of the order.

Solve the system of linear equations for

x

and

y

Express

4x + 9y

4(x + y) + 5y = \frac{7}{12}

. Substituting

x + y = \frac{1}{12}

gives

4\left(\frac{1}{12}\right) + 5y = \frac{7}{12} \implies \frac{4}{12} + 5y = \frac{7}{12} \implies 5y = \frac{3}{12} = \frac{1}{4} \implies y = \frac{1}{20}

. Subsequently,

x = \frac{1}{12} - \frac{1}{20} = \frac{5 - 3}{60} = \frac{2}{60} = \frac{1}{30}

Determines the individual work rates of Machine X and Machine Y.

Calculate the work rate of Machine Z.

Rate of Machine Z =

1.5 \times x = 1.5 \times \frac{1}{30} = \frac{3}{2} \times \frac{1}{30} = \frac{1}{20}

Machine Z is

50\%

faster than Machine X.

Calculate the combined rate and total time for Machine Y and Machine Z.

Combined rate =

y + z = \frac{1}{20} + \frac{1}{20} = \frac{2}{20} = \frac{1}{10}

. Total time =

\frac{1}{\frac{1}{10}} = 10

hours.

The inverse of the combined rate gives the total time required to finish one job.

Key Concept

Work Rates and Simultaneous Linear Modeling

Estimated Time:2m 30s

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