Answer the question below based on the given information.

Two containers of solutions, A and B, contain alcohol and water in the ratio of 5 : 4 and 2 : 1 respectively. A portion equal to 20% of solution A and 50% of solution B is combined to create a third solution C. If the total quantity of alcohol present in solution C is 90 liters and the ratio of alcohol to water in solution C is 30 : 19, then determine the initial quantity of solution A.

The correct option is 360 liters.

Step 1: Understand the given ratios for Solution A and Solution B

In Solution A, the ratio of alcohol to water is 5 : 4.
This means that in any quantity taken from Solution A:
Fraction of alcohol = 5 / (5 + 4) = 5 / 9
Fraction of water = 4 / (5 + 4) = 4 / 9

In Solution B, the ratio of alcohol to water is 2 : 1.
This means that in any quantity taken from Solution B:
Fraction of alcohol = 2 / (2 + 1) = 2 / 3
Fraction of water = 1 / (2 + 1) = 1 / 3

Step 2: Define variables for the initial quantities

Let the initial quantity of Solution A be $A$ liters.
Let the initial quantity of Solution B be $B$ liters.

Step 3: Determine the amounts taken from A and B to form Solution C

Quantity taken from Solution A = 20% of $A$ = 0.2$A$
Quantity taken from Solution B = 50% of $B$ = 0.5$B$

Step 4: Calculate the alcohol and water contributed to Solution C

Alcohol from Solution A = (5 / 9) × 0.2$A$
Alcohol from Solution B = (2 / 3) × 0.5$B$

Water from Solution A = (4 / 9) × 0.2$A$
Water from Solution B = (1 / 3) × 0.5$B$

Step 5: Set up equations using the total alcohol and the alcohol-to-water ratio in Solution C

We are given that total alcohol in Solution C is 90 liters:

$\frac{5}{9}\times 0.2A+\frac{2}{3}\times 0.5B=90$

$\frac{1}{9}A+\frac{1}{3}B=90$

Multiplying the entire equation by 9 to clear fractions:

$A+3B=810$   --- (Equation 1)

Next, we use the given ratio of alcohol to water in Solution C, which is 30 : 19.
Since total alcohol is 90 liters, we can find the total water in Solution C:

$\frac{\text{Alcohol in C}}{\text{Water in C}}=\frac{30}{19}$

$\frac{90}{\text{Water in C}}=\frac{30}{19}$

Water in Solution C = (90 × 19) / 30 = 3 × 19 = 57 liters.

Now, write the equation for total water in Solution C:

$\frac{4}{9}\times 0.2A+\frac{1}{3}\times 0.5B=57$

$\frac{0.8}{9}A+\frac{0.5}{3}B=57$

Multiplying the entire equation by 9 to clear the fraction:

$0.8A+1.5B=513$   --- (Equation 2)

Step 6: Solve the system of linear equations

Multiply Equation 2 by 2:

$1.6A+3B=1026$   --- (Equation 3)

Subtract Equation 1 from Equation 3:

$(1.6A+3B)-(A+3B)=1026-810$

$0.6A=216$

$A=\frac{216}{0.6}=360$

Conclusion:
The initial quantity of solution A is 360 liters.