Question

Nine gallons are drawn from a cask full of wine; it is then filled with water, then nine gallons of the mixture is drawn and the cask is again filled with water. If the quantity of wine now in the cask to the quantity of water in it is in the ratio 16 is to 9, how much does the cask hold?

Answer 1

Hint: Assume a variable for the volume of the cask. Then, use the given information to relate to the volume of the cask then write the quantity of remaining wine and water in the terms of x and equate them to the given ratio. Then solve the equations to find the answer.

Complete step-by-step solution -
Let the volume of the cask full of wine be x gallons.
It is given that 9 gallons are drawn from a cask full of wine and it is then filled with water. Hence, now, the volume of wine in the cask is x – 9 and it has 9 gallons of water.
The ratio of wine to water in this mixture is x – 9: 9.
Next, another 9 gallons of this mixture is replaced with water. Let us find out how many gallons of wine is there in this 9 gallons of mixture.
Volume of wine in 9 gallons of mixture = \[\dfrac{{x - 9}}{x} \times 9\]
Now, this is replaced with water. Hence, we have:
Volume of wine in the final mixture = \[x - 9 - \left( {\dfrac{{x - 9}}{x} \times 9} \right)\]
Volume of wine in the final mixture = \[x - 9 - \left( {\dfrac{{9x - 81}}{x}} \right)\]
Volume of wine in the final mixture = \[\dfrac{{{x^2} - 9x - 9x + 81}}{x}\]
Volume of wine in the final mixture = \[\dfrac{{{x^2} - 18x + 81}}{x}\]
Volume of wine in the final mixture = \[\dfrac{{{{(x - 9)}^2}}}{x}.............(1)\]
Volume of water in the final mixture = \[9 + \left( {\dfrac{{x - 9}}{x} \times 9} \right)\]
Volume of water in the final mixture = \[9 + \left( {\dfrac{{9x - 81}}{x}} \right)\]
Volume of water in the final mixture = \[\dfrac{{9x + 9x - 81}}{x}\]
Volume of water in the final mixture = \[\dfrac{{18x - 81}}{x}.............(2)\]
The ratio of the quantity of wine to water is 16:9, hence, using equations (1) and (2), we have:
\[\dfrac{{\dfrac{{{{(x - 9)}^2}}}{x}}}{{\dfrac{{9x - 81}}{x}}} = \dfrac{{16}}{9}\]
Simplifying, we have:
\[\dfrac{{{{(x - 9)}^2}}}{{9(x - 9)}} = \dfrac{{16}}{9}\]
Canceling common terms, we have:
\[\dfrac{{x - 9}}{9} = \dfrac{{16}}{9}\]
Canceling 9 on both sides, we have:
\[x - 9 = 16\]
\[x = 16 + 9\]
\[x = 25\]
Hence, the cask holds 25 gallons of liquid.

Note: You can also solve by assuming the quantity of water or wine in the final mixture and go backward to find the relation between the variables and finally find the total volume of the cask.