SOLUTION: divide 224 into two parts such that when the larger part is divided by the smaller part, the quotient is 2 and the remainder is half of the smaller part. find the larger part.

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Question 417383: divide 224 into two parts such that when the larger part is divided by the smaller part, the quotient is 2 and the remainder is half of the smaller part. find the larger part.
Found 2 solutions by ewatrrr, htmentor:
Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!


Hi

divide 224 into two parts

Let x and (224-x) represent the smaller and larger part respectively

Question states***

  %28224-x%29%2Fx+=+2+%2B+%281%2F2%29x

Solving for x

    2(224-x) = 4x + x^2   |Multiplying thru by 2x

    x^2 +6x -448 = 0 

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+

x+=+%28-6+%2B-+sqrt%28+1828%29%29%2F%282%29+

x+=+-3+%2B-+sqrt%28+457%29

  x = 18.3776    | Tossing out negative solution

  CHECKING our Answer***

   205.6224/18.3776 = 11.1888 = 2 + (1/2)18.3776 = 2 + 9.1888

Answer by htmentor(1343) About Me  (Show Source):
You can put this solution on YOUR website!
Let the two numbers be x and y
Then we have x%2By=224 (1)
This step is a little tricky. We are told the quotient is 2 and the remainder is half the denominator (the smaller number). The expression can be written
+x%2Fy+=+%282y+%2B+%281%2F2%29y%29%2Fy (2).
This might be hard to follow, so let's use an example: 15%2F6+=+%2812%2B3%29%2F6 Separating the terms, we have 15%2F6+=+12%2F6+%2B+3%2F6.
This says that the quotient is 2 with a remainder of 3, which is half the denominator (the smaller part).
Solving for x in (2) gives x+=+y%2A%282%2B0.5%29 or x=2.5y.
Substituting into (1) gives 2.5y+%2B+y+=+224 or y=224%2F3.5 -> y=64.
Therefore x+=+224-64+=+160.
So the smaller part is 64 and the larger part is 160.
Check: 160/64 = 2, with a remainder of 160-2*64 = 160-128=32, which is half of 64.