If there are three sum of money P,Q and R so that P is the simple interest of Q and Q is the simple interest of R, rate % and time are same in each case, then the relation of P, Q and R is given by

Options

A. `P^(2)=QR`
B. `Q^(2)=PR`
C. `R^(2)=PQ`
D. `PQR=100`

Answer

To solve the problem, we need to establish the relationship between the three sums of money: P, Q, and R, based on the given conditions.

### Step-by-Step Solution:

1. **Understanding the Definitions**:
- Let P be the simple interest on Q.
- Let Q be the simple interest on R.
- The rate of interest (r) and time (t) are the same for both cases.

2. **Using the Simple Interest Formula**:
- The formula for simple interest is given by:
\[ \text{Simple Interest} = \frac{P \times r \times t}{100} \]
- For the first case where P is the simple interest on Q:
\[ P = \frac{Q \times r \times t}{100} \]
- For the second case where Q is the simple interest on R:
\[ Q = \frac{R \times r \times t}{100} \]

3. **Rearranging the Equations**:
- From the first equation, we can express Q in terms of P:
\[ Q = \frac{P \times 100}{r \times t} \]
- From the second equation, we can express R in terms of Q:
\[ R = \frac{Q \times 100}{r \times t} \]

4. **Substituting Q in Terms of P**:
- Substitute the expression for Q into the equation for R:
\[ R = \frac{\left(\frac{P \times 100}{r \times t}\right) \times 100}{r \times t} \]
- Simplifying this gives:
\[ R = \frac{P \times 10000}{(r \times t)^2} \]

5. **Finding the Relationship Between P, Q, and R**:
- From the two equations:
\[ P = \frac{Q \times r \times t}{100} \]
\[ Q = \frac{R \times r \times t}{100} \]
- We can derive that:
\[ \frac{P}{Q} = \frac{Q}{R} \]
- Cross-multiplying gives:
\[ P \times R = Q^2 \]
- Thus, the relationship between P, Q, and R can be expressed as:
\[ Q^2 = P \times R \]

### Final Result:
The relation between P, Q, and R is:
\[ Q^2 = P \times R \]