Factory M has printed 650 books and prints 50 new books every week. Factory N has printed 200 books and prints 100 books every week. How many weeks (w) will it be before Factory N has printed as many books (b) as Factory M?
[tex]
\begin{array}{c}
b=50 w+650 \\
b=100 w+200
\end{array}
[/tex]
weeks
Answer (2)
Set up equations for the number of books each factory prints: b = 50 w + 650 and b = 100 w + 200 .
Equate the two expressions: 50 w + 650 = 100 w + 200 .
Solve for w : 50 w = 450 .
Find the number of weeks: w = 9 .
Explanation
Problem Analysis Let's analyze the problem. We have two factories, M and N, printing books. Factory M starts with 650 books and prints 50 more each week. Factory N starts with 200 books and prints 100 more each week. We need to find out after how many weeks the number of books printed by both factories will be the same.
Setting up the Equations Let b be the number of books and w be the number of weeks. We can write two equations:
For Factory M: b = 50 w + 650 For Factory N: b = 100 w + 200
Equating the Number of Books To find when the number of books is the same, we set the two equations equal to each other:
50 w + 650 = 100 w + 200
Solving for Weeks Now, let's solve for w :
Subtract 50 w from both sides: 650 = 50 w + 200
Subtract 200 from both sides: 450 = 50 w
Divide both sides by 50: w = 50 450 = 9
Final Answer So, it will take 9 weeks for Factory N to have printed as many books as Factory M.
Examples
Imagine you are managing two competing businesses. One starts with more customers but grows slowly, while the other starts with fewer customers but grows faster. This problem helps you determine when the second business will catch up to the first in terms of customer base. This type of problem is applicable in various scenarios such as comparing investment growth, analyzing market share, or even planning resource allocation.
Factory N will take 9 weeks to print as many books as Factory M. This is determined by setting up equations for the number of books printed and solving for the weeks needed. By equating the two expressions, we find when their outputs become equal.
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