If CI on some amount for 2 years at `16 (2)/(3)%` rate is 520 then find the amount?

Options

A. ₹ 1740
B. ₹ 1640
C. ₹ 1440
D. ₹ 1540

Answer

To solve the problem of finding the principal amount based on the compound interest (CI) earned over 2 years at a rate of \(16 \frac{2}{3}\%\), we can follow these steps:

### Step 1: Convert the Rate to a Fraction
The given rate is \(16 \frac{2}{3}\%\). To convert this to a fraction:
\[ 16 \frac{2}{3} = \frac{50}{3} \% \]
This means the rate in decimal form is:
\[ \frac{50}{3 \times 100} = \frac{50}{300} = \frac{1}{6} \]

### Step 2: Understand the Compound Interest Formula
The formula for compound interest for 2 years is given by:
\[ A = P \left(1 + r\right)^n \]
Where:
- \(A\) is the amount after time \(n\),
- \(P\) is the principal amount,
- \(r\) is the rate of interest per period,
- \(n\) is the number of periods.

### Step 3: Set Up the Equation
We know that the CI for 2 years is \(520\). The amount \(A\) can be expressed as:
\[ A = P + CI \]
Thus, we can write:
\[ A = P + 520 \]

### Step 4: Substitute into the Compound Interest Formula
Substituting \(A\) into the compound interest formula:
\[ P + 520 = P \left(1 + \frac{1}{6}\right)^2 \]
Calculating \(1 + \frac{1}{6}\):
\[ 1 + \frac{1}{6} = \frac{7}{6} \]
Now, squaring it:
\[ \left(\frac{7}{6}\right)^2 = \frac{49}{36} \]

### Step 5: Substitute Back
Now we substitute back into the equation:
\[ P + 520 = P \cdot \frac{49}{36} \]

### Step 6: Rearranging the Equation
Rearranging gives:
\[ P \cdot \frac{49}{36} - P = 520 \]
Factoring out \(P\):
\[ P \left(\frac{49}{36} - 1\right) = 520 \]
Calculating \(\frac{49}{36} - 1\):
\[ \frac{49}{36} - 1 = \frac{49 - 36}{36} = \frac{13}{36} \]
Thus, we have:
\[ P \cdot \frac{13}{36} = 520 \]

### Step 7: Solve for \(P\)
To find \(P\), multiply both sides by \(\frac{36}{13}\):
\[ P = 520 \cdot \frac{36}{13} \]
Calculating this gives:
\[ P = 520 \cdot 2.769 = 1400 \]

### Step 8: Find the Amount
Now, substituting \(P\) back to find \(A\):
\[ A = P + 520 = 1400 + 520 = 1920 \]

### Final Answer
The amount is \(1920\).