Solution:

For a mixture of \( n_1 \) moles of a monatomic gas and \( n_2 \) moles of a diatomic gas, the ratio \( \gamma = \frac{C_p}{C_v} \) of the mixture is given by:

\[
\gamma = \frac{\text{Total } C_p}{\text{Total } C_v}
\]

1. Molar heat capacities:
- For a monatomic gas: \( C_{v, \text{mono}} = \frac{3}{2} R \) and \( C_{p, \text{mono}} = \frac{5}{2} R \).
- For a diatomic gas: \( C_{v, \text{di}} = \frac{5}{2} R \) and \( C_{p, \text{di}} = \frac{7}{2} R \).

2. Total heat capacities:
- Total \( C_v = n_1 \cdot \frac{3}{2} R + n_2 \cdot \frac{5}{2} R \).
- Total \( C_p = n_1 \cdot \frac{5}{2} R + n_2 \cdot \frac{7}{2} R \).

3. Given \( \gamma = 1.5 \):

\[
\frac{C_p}{C_v} = \frac{n_1 \cdot \frac{5}{2} R + n_2 \cdot \frac{7}{2} R}{n_1 \cdot \frac{3}{2} R + n_2 \cdot \frac{5}{2} R} = 1.5
\]

4. Simplify by canceling \( R \) and multiplying through by 2:

\[
\frac{5n_1 + 7n_2}{3n_1 + 5n_2} = 1.5
\]

5. Cross-multiply to solve for \( n_1 \) in terms of \( n_2 \):

\[
5n_1 + 7n_2 = 1.5 (3n_1 + 5n_2)
\]

\[
5n_1 + 7n_2 = 4.5n_1 + 7.5n_2
\]

6. Rearrange terms:

\[
0.5n_1 = 0.5n_2
\]

\[
n_1 = n_2
\]