In a liquid solution, there are x liters of ethanol and (x + 25) liters of water. If $\frac{5}{3}$ liters of water are added to the solution, then ethanol becomes 30% of the new mixture. Find the initial volume of the solution in liters.

The correct answer is 65 liters.

Step 1: Define the initial quantities of the mixture
Let the volume of ethanol in the initial solution be x liters.
The volume of water in the initial solution is given as (x + 25) liters.
Therefore, the initial total volume of the liquid solution is:

$\text{Initial Total Volume}=x+(x+25)=2x+25\text{liters}$

Step 2: Account for the added water
When $\frac{5}{3}$ liters of water are added to the solution, the volume of ethanol remains unchanged at x liters, but the total volume of the mixture increases.
The new total volume of the mixture becomes:

$\text{New Total Volume}=2x+25+\frac{5}{3}$

Combining the constant terms:

$25+\frac{5}{3}=\frac{75+5}{3}=\frac{80}{3}$

So, the new total volume of the mixture is:

$\text{New Total Volume}=2x+\frac{80}{3}\text{liters}$

Step 3: Formulate the equation using the given percentage
It is given that ethanol becomes 30% of the new mixture.
Expressing 30% as a fraction, we get $\frac{30}{100}=\frac{3}{10}$.
Setting up the ratio of ethanol volume to new total volume:

$\frac{x}{2x+\frac{80}{3}}=\frac{3}{10}$

Step 4: Solve for x
Cross-multiply to solve the equation:

$10x=3\times \left(2x+\frac{80}{3}\right)$

Expand the right side:

$10x=6x+80$

Subtract 6x from both sides:

$4x=80$

$x=20$

Step 5: Calculate the initial volume of the solution
Substitute x = 20 back into the expression for the initial total volume:

$\text{Initial Volume}=2x+25$

$\text{Initial Volume}=2\left(20\right)+25=40+25=65\text{liters}$

Hence, the initial volume of the solution is 65 liters.