Four sums p, q, r and s are such that p is interest of q, qis interest of r and r is interest of k. If the time is same then find the value of P: K.

Options

A. `q^(3):r^(3)`
B. `r^(3):q^(3)`
C. `q^(2):r^(2)`
D. `r^(2):q`

Answer

To solve the problem, we need to establish relationships between the sums p, q, r, and k based on the information given. We know that:

1. \( p \) is the interest on \( q \)
2. \( q \) is the interest on \( r \)
3. \( r \) is the interest on \( k \)

Since the time is the same for all, we can use the formula for simple interest:

\[ \text{Simple Interest (SI)} = \frac{P \times R \times T}{100} \]

Where:
- \( P \) = Principal amount
- \( R \) = Rate of interest
- \( T \) = Time

### Step 1: Write the equations based on the relationships

1. For \( p \) as the interest of \( q \):
\[ p = \frac{q \times R \times T}{100} \quad \text{(1)} \]

2. For \( q \) as the interest of \( r \):
\[ q = \frac{r \times R \times T}{100} \quad \text{(2)} \]

3. For \( r \) as the interest of \( k \):
\[ r = \frac{k \times R \times T}{100} \quad \text{(3)} \]

### Step 2: Express \( p \), \( q \), and \( r \) in terms of \( k \)

From equation (3):
\[ r = \frac{k \times R \times T}{100} \implies k = \frac{r \times 100}{R \times T} \quad \text{(4)} \]

Substituting equation (4) into equation (2):
\[ q = \frac{\left(\frac{k \times R \times T}{100}\right) \times R \times T}{100} \]
\[ q = \frac{k \times R^2 \times T^2}{100^2} \quad \text{(5)} \]

Now substituting equation (5) into equation (1):
\[ p = \frac{\left(\frac{k \times R^2 \times T^2}{100^2}\right) \times R \times T}{100} \]
\[ p = \frac{k \times R^3 \times T^3}{100^3} \quad \text{(6)} \]

### Step 3: Find the ratio \( \frac{p}{k} \)

Now we can find the ratio \( \frac{p}{k} \):
\[ \frac{p}{k} = \frac{\frac{k \times R^3 \times T^3}{100^3}}{k} \]
\[ \frac{p}{k} = \frac{R^3 \times T^3}{100^3} \]

### Step 4: Conclusion

Thus, the ratio \( \frac{p}{k} \) is given by:
\[ \frac{p}{k} = \frac{R^3 \times T^3}{100^3} \]

### Final Result

The value of \( P:K \) is \( R^3:T^3 \).

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