Question 1203224
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Answer: <font color=red size=4>51 filing cabinets</font>
This is the starting amount for the regional office.


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How to get this answer:


The regional office has 1/4 the amount of the central office.
This must mean the central office has 4 times as many cabinets.


x = number of cabinets in regional office
4x = number of cabinets in central office
These are the starting amounts.
x is some positive integer.


The central office gives 9 cabinets
The amount 4x drops to 4x-9
Meanwhile, the amount x increases to x+9


We can organize the data as such
<table border = "1" cellpadding = "5"><tr><td></td><td>Regional</td><td>Central</td></tr><tr><td>Old</td><td>x</td><td>4x</td></tr><tr><td>New</td><td>x+9</td><td>4x-9</td></tr></table>


We are told that "<font color=blue>the regional office will have 4/13 as many as the number that remains with the central office</font>"


So,
regional = (4/13)*(central)
x+9 = (4/13)*(4x-9)
13(x+9) = 4(4x-9)
13x+117 = 16x-36
13x-16x = -36-117
-3x = -153
x = -153/(-3)
x = <font color=red>51</font>
The regional office starts with 51 cabinets.
The central office starts with 4x = 4*51 = 204 cabinets.



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Check:


If the central office sends 9 cabinets, then we'll have this table
<table border = "1" cellpadding = "5"><tr><td></td><td>Regional</td><td>Central</td></tr><tr><td>Old</td><td>51</td><td>204</td></tr><tr><td>New</td><td>60</td><td>195</td></tr></table>
Then take the ratio of the values in the bottom row.
60/195 = (15*4)/(15*13) = 4/13
This confirms that the regional office ends up with 4/13 of the amount of cabinets compared to the central office.
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