Three machines—X, Y, and Z—produce widgets at constant rates. The ratio of the production … — ACT

Three machines— $X$ , $Y$ , and $Z$ —produce widgets at constant rates. The ratio of the production rate of Machine $X$ to Machine $Y$ to Machine $Z$ is $3:4:6$ , respectively. Working together at their constant rates, the three machines can complete a standard production order in exactly $8$ hours. A new order of the same size is started with all three machines working together. After $3$ hours, Machine $Z$ breaks down and stops working, and the production rate of Machine $Y$ decreases by $25\%$ . If Machine $X$ and the slowed Machine $Y$ continue working to complete the order, what is the total number of hours required to complete this entire second order?

$13\frac{5}{6}$ Answer
B
$10\frac{5}{6}$
C
$8$
D
$11\frac{1}{3}$
E
$20\frac{1}{3}$

Answer

13\frac{5}{6}

hours

The correct answer is

13\frac{5}{6}

hours. By translating the ratios into algebraic rates, we determine that the total order size is

104r

widgets. In the first

3

hours,

39r

widgets are produced, leaving

65r

widgets. After the breakdown and slowdown, the combined rate of Machines

X

and

Y

3r + 3r = 6r

widgets per hour. The remaining work requires

\frac{65r}{6r} = 10\frac{5}{6}

hours, making the total duration

3 + 10\frac{5}{6} = 13\frac{5}{6}

hours.

Step-by-Step Solution

Define the rate of each machine and find the total work required for the order.

Let the rates of Machine

X

Y

, and

Z

3r

4r

, and

6r

widgets per hour, respectively. The combined rate is

3r + 4r + 6r = 13r

widgets per hour. The total work

W

13r \times 8 = 104r

widgets.

Establishing rates using the given ratios allows us to compute the size of the job in terms of rate units.

Calculate the work completed in the first 3 hours and find the remaining work.

Work completed in the first

3

hours is

13r \times 3 = 39r

widgets. The remaining work is

104r - 39r = 65r

widgets.

We must subtract the work done before the breakdown to find the remaining workload.

Determine the new rates after Machine

Z

breaks down and Machine

Y

slows down.

Machine

Z

's rate becomes

0

. Machine

Y

's new rate is

4r \times (1 - 0.25) = 3r

widgets per hour. The new combined rate for Machines

X

and

Y

3r + 3r = 6r

widgets per hour.

We apply the rate changes to calculate the rate at which the remaining work will be completed.

Calculate the time needed to complete the remaining work and add the initial time.

The remaining work takes

\frac{65r}{6r} = 10\frac{5}{6}

hours. The total time for the entire order is

3 + 10\frac{5}{6} = 13\frac{5}{6}

hours.

Dividing the remaining work by the new combined rate gives the remaining time, which we add to the initial 3 hours to get the total time.

Key Concept

Rate and ratio application to multi-stage work problems

Question

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