A sum of money is paid back in three annual installments of ₹ 1372 each, allowing `16 (2)/(3)%` compound interst compounded annually. The sum borrowed was:

Options

A. ₹ 3100
B. ₹ 3148
C. ₹ 3048
D. ₹ 3248

Answer

To find the sum borrowed that is paid back in three annual installments of ₹1372 each, with a compound interest rate of \(16 \frac{2}{3}\%\) (or \(\frac{50}{3}\%\)), we can follow these steps:

### Step 1: Convert the Interest Rate
The interest rate is given as \(16 \frac{2}{3}\%\). We convert this to a fraction:
\[ r = 16 \frac{2}{3}\% = \frac{50}{3}\% \]

### Step 2: Calculate the Present Value of Each Installment
The formula for the present value (PV) of an installment is:
\[ PV = \frac{R}{(1 + \frac{r}{100})^t} \]
where \(R\) is the installment amount, \(r\) is the interest rate, and \(t\) is the time in years.

#### For the first installment (t = 1):
\[ PV_1 = \frac{1372}{(1 + \frac{50}{3 \times 100})^1} = \frac{1372}{(1 + \frac{50}{300})} = \frac{1372}{\frac{7}{6}} = 1372 \times \frac{6}{7} \]

#### For the second installment (t = 2):
\[ PV_2 = \frac{1372}{(1 + \frac{50}{3 \times 100})^2} = \frac{1372}{(1 + \frac{50}{300})^2} = \frac{1372}{\left(\frac{7}{6}\right)^2} = 1372 \times \frac{36}{49} \]

#### For the third installment (t = 3):
\[ PV_3 = \frac{1372}{(1 + \frac{50}{3 \times 100})^3} = \frac{1372}{(1 + \frac{50}{300})^3} = \frac{1372}{\left(\frac{7}{6}\right)^3} = 1372 \times \frac{216}{343} \]

### Step 3: Calculate the Total Present Value
Now, we sum the present values of all three installments:
\[ \text{Total PV} = PV_1 + PV_2 + PV_3 \]
Substituting the values:
\[ \text{Total PV} = 1372 \times \frac{6}{7} + 1372 \times \frac{36}{49} + 1372 \times \frac{216}{343} \]

### Step 4: Factor Out the Common Term
Factoring out \(1372\):
\[ \text{Total PV} = 1372 \left(\frac{6}{7} + \frac{36}{49} + \frac{216}{343}\right) \]

### Step 5: Find a Common Denominator
The common denominator for \(7\), \(49\), and \(343\) is \(343\):
\[ \text{Total PV} = 1372 \left(\frac{6 \times 49}{343} + \frac{36 \times 7}{343} + \frac{216}{343}\right) \]
Calculating each term:
\[ = 1372 \left(\frac{294 + 252 + 216}{343}\right) = 1372 \left(\frac{762}{343}\right) \]

### Step 6: Final Calculation
Now, we calculate the total present value:
\[ = 1372 \times \frac{762}{343} = \frac{1041964}{343} \approx 3048 \]

Thus, the sum borrowed was approximately ₹3048.