A certain sample of cuprous sulphide is found to have the composition Cu1.92S1.00 because of incorporation of Cu2+ and Cu+ ions in the crystal. Then, the ratio of Cu2+ and Cu+ ions is: (a) 0.08: 1.00
(b) 1:23
(c) 1:24
(d) 1:1

To solve the problem of determining the ratio of \(\text{Cu}^{2+}\) to \(\text{Cu}^+\) ions in the sample of cuprous sulfide with the formula \(\text{Cu}_{1.92}\text{S}_{1.00}\), follow these steps:

### 1. **Understand the Composition:**

The formula \(\text{Cu}_{1.92}\text{S}_{1.00}\) indicates that the sample contains 1.92 copper (Cu) atoms for every 1 sulfur (S) atom.

### 2. **Determine the Charge Balance:**

In cuprous sulfide, the sulfur is in the \(\text{S}^{2-}\) oxidation state. To maintain electrical neutrality, the sum of the charges of copper ions must balance the charge of sulfur.

### 3. **Let \( x \) be the number of \(\text{Cu}^{2+}\) ions and \( y \) be the number of \(\text{Cu}^+\) ions.**

Since the total number of copper ions is given as 1.92, we have:

\[ x + y = 1.92 \]

The total charge must balance the charge of sulfur. Each \(\text{S}^{2-}\) ion contributes a charge of \(-2\), so for 1 sulfur atom:

\[ 2x + y = 2 \]

### 4. **Solve the System of Equations:**

We have two equations:

1. \( x + y = 1.92 \)
2. \( 2x + y = 2 \)

Subtract the first equation from the second to eliminate \( y \):

\[ (2x + y) - (x + y) = 2 - 1.92 \]

This simplifies to:

\[ x = 0.08 \]

Now substitute \( x = 0.08 \) into the first equation:

\[ 0.08 + y = 1.92 \]

Solve for \( y \):

\[ y = 1.92 - 0.08 = 1.84 \]

### 5. **Determine the Ratio:**

The ratio of \(\text{Cu}^{2+}\) to \(\text{Cu}^+\) is:

\[ \text{Ratio} = \frac{x}{y} = \frac{0.08}{1.84} \]

To simplify:

\[ \frac{0.08}{1.84} = \frac{8}{184} = \frac{1}{23} \]

### **Conclusion:**

The ratio of \(\text{Cu}^{2+}\) to \(\text{Cu}^+\) ions is \(1:23\), so the correct answer is:

**(b) 1:23**