Four gallons are drawn from a case full of wine. It is then filled with water. Four gallons of mixture are again drawn and the cask is re-filled with water. The ratio of the quantity of wine now left in the cask to that of the mixture in it is 36:49. How much does the cask hold?a)30 gallonsb)25 gallonsc)35 gallonsd)28 gallonsCorrect answer is option 'D'. Can you explain this answer?

Question

Answer

Given information:
- Four gallons are drawn from a case full of wine.
- It is then filled with water.
- Four gallons of mixture are again drawn and the cask is re-filled with water.
- The ratio of the quantity of wine now left in the cask to that of the mixture in it is 36:49.

To find:
- How much does the cask hold?

Solution:
Let's assume that the cask initially holds 'x' gallons of wine.

After drawing 4 gallons of wine, the amount of wine left in the cask = x - 4 gallons.
The cask is then filled with water, so the total amount of liquid in the cask = x gallons.

Now, 4 gallons of mixture are drawn from the cask. This mixture contains some wine and some water in it. Let's assume that the amount of wine in this mixture is 'y' gallons.

So, the amount of water in this mixture = (4 - y) gallons.
When this mixture is drawn from the cask, the amount of wine left in the cask = (x - 4) - y gallons.
The cask is then re-filled with water, so the total amount of liquid in the cask = x gallons.

According to the given information, the ratio of the wine left in the cask to the mixture in it is 36:49.
So, we can write the following equation:

(x - 4 - y) : (4 - y) = 36 : 49

Cross-multiplying this equation, we get:

49(x - 4 - y) = 36(4 - y)

Simplifying this equation, we get:

49x - 196 - 49y = 144 - 36y

49x + 13y = 340 ...........(1)

We also know that the total amount of liquid in the cask after the second filling is x gallons. So, we can write another equation:

x = y + 4 ...........(2)

Substituting the value of 'y' from equation (2) into equation (1), we get:

49x + 13(y+4) = 340

49x + 13y + 52 = 340

49x + 13y = 288

Dividing both sides by 13, we get:

3x + y = 22 ...........(3)

We have three equations (1), (2), and (3) in three variables (x, y, and z). We can solve these equations to find the values of 'x', 'y', and 'z'.

Multiplying equation (3) by 13 and subtracting it from equation (1), we get:

36x = 52

x = 52/36 = 13/9

Substituting the value of 'x' in equation (2), we get:

y + 4 = 13/9

y = 13/9 - 4 = 5/9

So, the amount of wine in the cask initially = x = 13/9 gallons
The amount of wine in the mixture drawn first time = y = 5/9 gallons
The amount of water in the mixture drawn first time = 4 - y = 31/9 gallons
The amount of water in the cask initially =

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