SOLUTION: A certain regional office has ¼ as many filing cabinets as its central office. If the central office gives 9 filing cabinets to its regional office, then the regional office will

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Question 1203224: A certain regional office has ¼ as many filing cabinets as its central office. If the central office gives 9 filing cabinets to its regional office, then the regional office will have 4/13 as many as the number that remains with the central office. How many cabinets does the regional office have?
Answer by math_tutor2020(3817) (Show Source):
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Answer: 51 filing cabinets
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

Regional Central
Old x 4x
New x+9 4x-9

We are told that "the regional office will have 4/13 as many as the number that remains with the central office"

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 = 51
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

Regional Central
Old 51 204
New 60 195

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.