p,q,r,w are 4 amounts such that p is S.I. of q.q is S.I. of r and r is S.I. of w. If the time & rate of intereset is equal to all three case find the ratio of p and w ?

Answer

To solve the problem step by step, we will use the formula for Simple Interest (SI), which is given by:

\[ \text{SI} = \frac{P \times R \times T}{100} \]

Where:
- P = Principal amount
- R = Rate of interest
- T = Time period

Given that:
- \( P \) is the simple interest on \( Q \)
- \( Q \) is the simple interest on \( R \)
- \( R \) is the simple interest on \( W \)
- The time and rate of interest are equal in all three cases.

### Step 1: Express \( P \) in terms of \( Q \)

Since \( P \) is the simple interest on \( Q \):
\[ P = \frac{Q \times R \times T}{100} \]

### Step 2: Express \( Q \) in terms of \( R \)

Since \( Q \) is the simple interest on \( R \):
\[ Q = \frac{R \times R \times T}{100} \]

### Step 3: Express \( R \) in terms of \( W \)

Since \( R \) is the simple interest on \( W \):
\[ R = \frac{W \times R \times T}{100} \]

### Step 4: Substitute \( Q \) and \( R \) into the equation for \( P \)

From Step 2, we have:
\[ Q = \frac{R^2 \times T}{100} \]

From Step 3, we can express \( R \):
\[ R = \frac{W \times R \times T}{100} \]

### Step 5: Find the ratio of \( P \) and \( W \)

We need to find the ratio \( \frac{P}{W} \). We can substitute the expressions we derived for \( Q \) and \( R \) back into the equation for \( P \).

1. Substitute \( R \) from Step 3 into the equation for \( Q \):
\[ Q = \frac{( \frac{W \times R \times T}{100} )^2 \times T}{100} \]

2. Now substitute this expression for \( Q \) back into the equation for \( P \):
\[ P = \frac{( \frac{( \frac{W \times R \times T}{100} )^2 \times T}{100}) \times R \times T}{100} \]

3. Simplifying this will lead to a relationship between \( P \) and \( W \).

After simplification, we find:
\[ \frac{P}{W} = \frac{Q^3}{R^3} \]

### Final Result

Thus, the ratio of \( P \) and \( W \) is:
\[ \frac{P}{W} = \frac{Q^3}{R^3} \]