If x,y z are three sum of money such that y is the simple interset on x and z is the simple interst on y for the same time and at the same rate of interset, then we have

Options

A. `z^(2) = xy`
B. xyz = 1
C. `x^(2) = yz`
D. `y^(2) = zx`

Answer

To solve the problem step by step, we need to understand the relationship between the sums of money \( x \), \( y \), and \( z \) based on the simple interest formula.

### Step-by-Step Solution

1. **Understanding the Problem**:
- We have three sums of money: \( x \), \( y \), and \( z \).
- \( y \) is the simple interest on \( x \).
- \( z \) is the simple interest on \( y \).
- The time period and rate of interest are the same for both cases.

2. **Using the Simple Interest Formula**:
- The formula for simple interest (SI) is given by:
\[ \text{SI} = \frac{P \cdot R \cdot T}{100} \]
- Where \( P \) is the principal amount, \( R \) is the rate of interest, and \( T \) is the time period.

3. **Setting Up the Equations**:
- For the first case, where \( y \) is the simple interest on \( x \):
\[ y = \frac{x \cdot r \cdot t}{100} \quad \text{(Equation 1)} \]
- For the second case, where \( z \) is the simple interest on \( y \):
\[ z = \frac{y \cdot r \cdot t}{100} \quad \text{(Equation 2)} \]

4. **Dividing the Equations**:
- Now, we divide Equation 1 by Equation 2:
\[ \frac{y}{z} = \frac{\frac{x \cdot r \cdot t}{100}}{\frac{y \cdot r \cdot t}{100}} \]
- Simplifying this gives:
\[ \frac{y}{z} = \frac{x}{y} \]

5. **Cross Multiplying**:
- Cross multiplying gives us:
\[ y^2 = z \cdot x \]

6. **Final Result**:
- Thus, we have derived the relationship:
\[ y^2 = z \cdot x \]

### Conclusion
The correct relationship derived from the problem is \( y^2 = z \cdot x \), which corresponds to option 4.