Find that principal amount on which the difference of interest for 9 months yearly and quarterly is Rs. 18.30 at 20% per annum.

Answer

To find the principal amount on which the difference of interest for 9 months yearly and quarterly is Rs. 18.30 at 20% per annum, we will follow these steps:

### Step 1: Convert the time period from months to years
Given that the time period is 9 months, we need to convert this into years for our calculations.

\[ \text{Time in years} = \frac{9 \text{ months}}{12 \text{ months/year}} = 0.75 \text{ years} \]

**Hint:** Remember that there are 12 months in a year, so to convert months to years, divide the number of months by 12.

### Step 2: Use the formula for calculating interest
The difference in interest for the same principal amount when calculated yearly and quarterly can be expressed as:

\[ \text{Difference in Interest} = I_{yearly} - I_{quarterly} \]

Where:
- \( I_{yearly} = P \times R \times T \)
- \( I_{quarterly} = P \times \left(\frac{R}{4}\right) \times \left(\frac{T}{4}\right) \)

Given that the difference in interest is Rs. 18.30, we can set up the equation:

\[ I_{yearly} - I_{quarterly} = 18.30 \]

### Step 3: Substitute the values into the interest formulas
Substituting the values into the interest formulas, we have:

\[ P \times (R \times T) - P \times \left(\frac{R}{4} \times \frac{T}{4}\right) = 18.30 \]

Substituting \( R = 20\% = \frac{20}{100} = 0.20 \) and \( T = 0.75 \):

\[ P \times (0.20 \times 0.75) - P \times \left(\frac{0.20}{4} \times \frac{0.75}{4}\right) = 18.30 \]

### Step 4: Calculate the interest values
Calculating \( I_{yearly} \):

\[ I_{yearly} = P \times 0.20 \times 0.75 = P \times 0.15 \]

Calculating \( I_{quarterly} \):

\[ I_{quarterly} = P \times \left(\frac{0.20}{4} \times \frac{0.75}{4}\right) = P \times \left(0.05 \times 0.1875\right) = P \times 0.009375 \]

### Step 5: Set up the equation for the difference
Now, substituting back into the difference equation:

\[ P \times 0.15 - P \times 0.009375 = 18.30 \]

Factoring out \( P \):

\[ P \times (0.15 - 0.009375) = 18.30 \]

Calculating \( 0.15 - 0.009375 \):

\[ 0.15 - 0.009375 = 0.140625 \]

So, we have:

\[ P \times 0.140625 = 18.30 \]

### Step 6: Solve for the principal \( P \)
Now, we can solve for \( P \):

\[ P = \frac{18.30}{0.140625} \]

Calculating \( P \):

\[ P \approx 130.0 \]

### Final Answer
The principal amount is approximately Rs. 130.00.

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