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Difficulty: Very hardLinear Equations in One Variable

Two commercial printing presses, Press A and Press B, operate at constant rates to print a total order of NN pages. Press A prints at a constant rate of xx pages per minute, while Press B prints at a constant rate that is 2525 pages per minute faster than Press A. Press A begins printing alone. After 2020 minutes, Press B is turned on, and both presses work simultaneously for an additional 3030 minutes. At that point, Press A stops, and Press B works alone for 1010 final minutes to complete the order. If Press B printed exactly 611\frac{6}{11} of the total number of pages in the order, what is the value of xx?

Answer: 50

Answer

The value of xx is 50.
To find xx, calculate the total time each press operated. Press A ran for 20 minutes alone plus 30 minutes with Press B, giving 50 minutes total at xx pages per minute (50x50x pages). Press B ran for 30 minutes with Press A plus 10 minutes alone, giving 40 minutes total at (x+25)(x + 25) pages per minute (40x+100040x + 1000 pages). The total order size is N=90x+1000N = 90x + 1000. Setting Press B's output equal to 611N\frac{6}{11}N yields 40x+1000=611(90x+1000)40x + 1000 = \frac{6}{11}(90x + 1000). Multiplying both sides by 11 gives 440x+11000=540x+6000440x + 11000 = 540x + 6000, which simplifies to 100x=5000100x = 5000, giving x=50x = 50.

Step-by-Step Solution

1
Determine total operating times and express pages printed by each press in terms of xx
Press A printed 50x50x pages; Press B printed 40(x+25)=40x+100040(x + 25) = 40x + 1000 pages.
Press A operated for 20 minutes alone plus 30 minutes together (50 minutes total). Press B operated for 30 minutes together plus 10 minutes alone (40 minutes total).
2
Write the expression for total pages NN
N=50x+(40x+1000)=90x+1000N = 50x + (40x + 1000) = 90x + 1000
The total pages in the order is the sum of the pages printed by Press A and Press B.
3
Formulate the linear equation in one variable using the given ratio
40x+1000=611(90x+1000)40x + 1000 = \frac{6}{11}(90x + 1000)
Press B printed exactly 611\frac{6}{11} of the total pages NN.
4
Solve the linear equation for xx
x=50x = 50
Multiplying both sides by 11 clears the fraction to give 440x+11000=540x+6000440x + 11000 = 540x + 6000, which simplifies to 100x=5000100x = 5000.

Key Concept

Formulating and solving linear equations in one variable from multi-step rate and work scenarios.
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Topics

Linear Equations in One Variable

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