Let the initial values of the variables be $x_0, y_0, z_0$.
Their initial ratio is $x_0: y_0: z_0=3: 24: 100$.
This means we can write:

$$
\(\begin{aligned}
& x_0=3 k \\
& y_0=24 k \\
& z_0=100 k
\end{aligned}\)
$$

for some constant $k$.

Now, let the new values of the variables be $x_1, y_1, z_1$.
Condition 1: Ratio of $x$ to $z$ is doubled.
Initial ratio of $x$ to $\(z: \frac{x_0}{z_0}=\frac{3 k}{100 k}=\frac{3}{100}\)$.
The new ratio of $x$ to $z$ is doubled: $\(\frac{x_1}{z_1}=2 \times \frac{3}{100}=\frac{6}{100}=\frac{3}{50}\)$.
Condition 2: Ratio of $y$ to $z$ is halved.
Initial ratio of $y$ to $\(z: \frac{y_0}{z_0}=\frac{24 k}{100 k}=\frac{24}{100}=\frac{6}{25}\)$.
The new ratio of $y$ to $z$ is halved: $\(\frac{y_1}{z_1}=\frac{1}{2} \times \frac{24}{100}=\frac{12}{100}=\frac{3}{25}\)$.
Given Information:
The new value of $x$ is 24 . So, $\(x_1=24\)$.

Goal: Find the new value of $y$, i.e., $y_1$.

From Condition 1, we have $\(\frac{x_1}{z_1}=\frac{3}{50}\)$.
Substitute $\(x_1=24\)$ :

$$
\(\frac{24}{z_1}=\frac{3}{50}\)
$$


To solve for $z_1$ :

$$
\(\begin{aligned}
& 3 \cdot z_1=24 \cdot 50 \\
& z_1=\frac{24 \cdot 50}{3} \\
& z_1=8 \cdot 50 \\
& z_1=400
\end{aligned}\)
$$


Now that we have the new value of $\(z, z_1=400\)$, we can use Condition 2 to find $\(y_1\)$.
From Condition 2, we have $\(\frac{y_1}{z_1}=\frac{3}{25}\)$.
Substitute $z_1=400$ :

$$
\(\frac{y_1}{400}=\frac{3}{25}\)
$$


To solve for $y_1$ :

$$
\(\begin{aligned}
& y_1=\frac{3}{25} \times 400 \\
& y_1=3 \times \frac{400}{25} \\
& y_1=3 \times 16 \\
& y_1=48
\end{aligned}\)
$$


So, the new value of $y$ is 48 .