Problem Solving

gmatter2012 wrote:A tin is full of wine.Out of this tin, 4 gallons of wine are taken out and 4 gallons of water are added.After this 4 gallons of mixture were taken out and again 4 gallons of water were added.Now the ratio of wine and water in the tin becomes 36:13.Find the volume of tin.

Say, the volume of the tin is x gallons.
Hence, initially there was x gallons of wine.

After first operation,

• There is (x - 4) gallons of wine and 4 gallons of water in the mixture.
  Hence, in x gallons there is (x - 4) gallons of wine.
  Hence, in 1 gallon there is (x - 4)/x gallons of wine

Therefore, in 4 gallons of the mixture there will be 4(x - 4)/x gallons of wine.
Similarly, in 4 gallons of the mixture there will be 4*4/x = 16/x gallons of water.
***
After second operation,

• There is [(x - 4) - 4(x - 4)/x] = (x - 4)(1 - 4/x) = (x - 4)(x - 4)/x = (x - 4)²/x gallons of wine and (4 - 16/x + 4) = (8 - 16/x) = (8x - 16)/x gallons of water in the mixture.
  
  Hence, ratio of wine and water = [(x - 4)²/x]/[(8x - 16)/x] = (x - 4)²/(8x - 16)

So, (x - 4)²/(8x - 16) = 36/13
--> 13(x - 4)² = 36*(8x - 16)
--> 13(x² - 8x + 16) = 288x - 36*16
--> 13x² - 104x + 13*16 = 288x - 36*16
--> 13x² - 392x + (13*16 + 36*16) = 0
--> 13x² - 392x + 16*(13 + 36) = 0
--> 13x² - 8*49x + 16*49 = 0
--> 13x² - 14*28x + 28*28 = 0
--> 13x² - 13*28x - 28x + 28*28 = 0
--> 13x(x - 28) - 28(x - 28) = 0
--> (x - 28)(13x - 28) = 0

Hence, either x = 28 or x = 28/13 = 2.(something)
Now, x must be greater than 4 as we are taking out 4 gallons from x gallons.

Therefore, only valid solution is x = 28
***
Hence volume of the tin is 28 gallons.

Note : The cumbersome algebra between the blue stars can be reduced to a great extent. Please see my post below.

Thanks for pointing it out.
Edited the reply.

As singhumesh noted above the calculation can be reduced by a large amount. I'm not editing my original post just explaining how it can be reduced.

After second operation,

• There is [(x - 4) - 4(x - 4)/x] = (x - 4)(1 - 4/x) = (x - 4)(x - 4)/x = (x - 4)²/x gallons of wine and volume of the mixture is x gallons
  Hence, ratio of wine to total volume = [(x - 4)²/x]/x = (x - 4)²/x²

Now, ratio of wine to water = 36/13
Hence, ratio of wine to total volume = 36/(36 + 13) = 36/49

So, (x - 4)²/x² = 36/49
--> [(x - 4)/x]² = (6/7)²
--> (x - 4)/x = 6/7
--> (1 - 4/x) = 6/7
--> 4/x = 1/7
--> x = 28