A sum of money amounts to Rs 10240 in 2 years at `6(2)/(3)%` per annum, compounded annually.
Find the sum.

Answer

To find the sum of money (the principal amount) that amounts to Rs 10,240 in 2 years at a compound interest rate of 6⅔% per annum, we can use the formula for compound interest:

\[ A = P \left(1 + \frac{r}{100}\right)^n \]

Where:
- \( A \) = the amount of money accumulated after n years, including interest.
- \( P \) = the principal amount (the initial sum of money).
- \( r \) = the annual interest rate (in percentage).
- \( n \) = the number of years the money is invested or borrowed.

### Step 1: Identify the given values
- Amount \( A = 10,240 \)
- Rate \( r = 6 \frac{2}{3} \% = \frac{20}{3} \% \) (convert mixed fraction to improper fraction)
- Time \( n = 2 \) years

### Step 2: Convert the interest rate to a decimal
To use the formula, we need to convert the percentage into a decimal:
\[ r = \frac{20}{3} \% = \frac{20}{3 \times 100} = \frac{1}{15} \]

### Step 3: Substitute the values into the compound interest formula
We need to rearrange the formula to find \( P \):
\[ P = \frac{A}{\left(1 + \frac{r}{100}\right)^n} \]

Substituting the values we have:
\[ P = \frac{10,240}{\left(1 + \frac{20}{300}\right)^2} \]

### Step 4: Simplify the expression inside the parentheses
First, calculate \( 1 + \frac{20}{300} \):
\[ 1 + \frac{20}{300} = 1 + \frac{1}{15} = \frac{15}{15} + \frac{1}{15} = \frac{16}{15} \]

### Step 5: Raise the result to the power of \( n \)
Now raise \( \frac{16}{15} \) to the power of 2:
\[ \left(\frac{16}{15}\right)^2 = \frac{256}{225} \]

### Step 6: Substitute back into the formula for \( P \)
Now substitute this back into the equation for \( P \):
\[ P = \frac{10,240}{\frac{256}{225}} \]

### Step 7: Simplify to find \( P \)
To divide by a fraction, multiply by its reciprocal:
\[ P = 10,240 \times \frac{225}{256} \]

Calculating this:
\[ P = \frac{10,240 \times 225}{256} \]

### Step 8: Calculate the final value of \( P \)
Now perform the multiplication and division:
\[ P = \frac{2,304,000}{256} = 9,000 \]

### Final Answer
The sum (principal amount) is Rs 9,000.

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