1. Set up degree of dissociation for both reactions

Let the initial amount of each reactant be the same, say $1$ mole, and let the common degree of dissociation be $\alpha$.

Since the problem states that the initial concentration and the degree of dissociation are equal for both reactants $X(g)$ and $A(g)$, we can use the same initial moles and same $\alpha$ for both systems.

2. Reaction (i):

$X(g) \rightleftharpoons Y(g)+Z(g), \qquad K_{p1}=3$

Initial moles:

$X=1,\quad Y=0,\quad Z=0$

At equilibrium:

$X=1-\alpha,\quad Y=\alpha,\quad Z=\alpha$

Total moles at equilibrium:

$n_{\text{tot}}=1-\alpha+\alpha+\alpha=1+\alpha$

If total pressure is $p_1$, then partial pressures are

$p_X=\frac{1-\alpha}{1+\alpha}p_1, \qquad p_Y=\frac{\alpha}{1+\alpha}p_1, \qquad p_Z=\frac{\alpha}{1+\alpha}p_1$

Now,

$K_{p1}=\frac{p_Y p_Z}{p_X}$

So,

$3=\frac{\left(\frac{\alpha}{1+\alpha}p_1\right)\left(\frac{\alpha}{1+\alpha}p_1\right)}{\left(\frac{1-\alpha}{1+\alpha}p_1\right)}$

Simplify:

$3=\frac{\alpha^2 p_1}{(1+\alpha)(1-\alpha)}$

$3=\frac{\alpha^2 p_1}{1-\alpha^2}$

Hence,

$p_1=3\frac{1-\alpha^2}{\alpha^2}$

3. Reaction (ii):

$A(g) \rightleftharpoons 2B(g), \qquad K_{p2}=1$

Initial moles:

$A=1,\quad B=0$

At equilibrium:

$A=1-\alpha,\quad B=2\alpha$

Total moles at equilibrium:

$n_{\text{tot}}=1-\alpha+2\alpha=1+\alpha$

If total pressure is $p_2$, then partial pressures are

$p_A=\frac{1-\alpha}{1+\alpha}p_2, \qquad p_B=\frac{2\alpha}{1+\alpha}p_2$

Now,

$K_{p2}=\frac{p_B^2}{p_A}$

So,

$1=\frac{\left(\frac{2\alpha}{1+\alpha}p_2\right)^2}{\left(\frac{1-\alpha}{1+\alpha}p_2\right)}$

Simplify:

$1=\frac{4\alpha^2 p_2}{(1+\alpha)(1-\alpha)}$

$1=\frac{4\alpha^2 p_2}{1-\alpha^2}$

Hence,

$p_2=\frac{1-\alpha^2}{4\alpha^2}$

4. Find the ratio $\dfrac{p_1}{p_2}$

$\frac{p_1}{p_2}=\frac{3\frac{1-\alpha^2}{\alpha^2}}{\frac{1-\alpha^2}{4\alpha^2}}$

Cancel common terms:

$\frac{p_1}{p_2}=3\times 4=12$

Thus,

$\frac{p_1}{p_2}=12:1$

So,

$x=12$

5. Comparison with stored answer

Derived answer: $12$

Stored correct answer: $12$

They match.