A container holds 528 liters of a mixture of acid and water in the ratio 7 : 5 respectively. From this mixture, 198 liters are taken out and replaced with (z + 22) liters of acid, so that the new ratio of acid to water becomes 9 : 5. Find the value of z.

To find the value of $z$, we can break down the problem step-by-step by analyzing the quantities of acid and water at each stage.

Step 1: Understand the Initial Mixture
The container initially holds $528$ liters of a mixture of acid and water in the ratio $7:5$.

Step 2: Calculate the Quantities After Removing 198 Liters
When we take out $198$ liters of the mixture, the remaining volume of the mixture in the container is:
$528-198=330\text{liters}$

The ratio of acid to water in this remaining $330$ liters remains unchanged at $7:5$.
The total number of parts in the ratio is:
$7+5=12\text{parts}$

We can now find the exact quantities of acid and water remaining in the container:

$\text{Remaining Acid}=330\times \frac{7}{12}=192.5\text{liters}$

$\text{Remaining Water}=330\times \frac{5}{12}=137.5\text{liters}$

Step 3: Set Up the Equation for the New Mixture
Next, $(z+22)$ liters of pure acid are added to the container. The quantity of water remains unchanged at $137.5$ liters.

The new quantity of acid is:
$\text{New Acid}=192.5+(z+22)=214.5+z\text{liters}$

According to the problem, the new ratio of acid to water is $9:5$. We can set up the proportion as follows:

$\frac{214.5+z}{137.5}=\frac{9}{5}$

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

$5\times (214.5+z)=9\times 137.5$

$1072.5+5z=1237.5$

Subtract $1072.5$ from both sides:

$5z=1237.5-1072.5$

$5z=165$

Divide by $5$:

$z=\frac{165}{5}$

$z=33$

Thus, the value of $z$ is $33$.