13th year compound interest 1024. Find the c.i. of 10th year if rate of interest is `14 (2)/(7) %`

Options

A. 443
B. 512
C. 343
D. 686

Answer

To solve the problem step by step, we will follow the given information and apply the formula for compound interest.

### Step 1: Identify the given values
- Amount at the end of 13 years (A) = 1024
- Rate of interest (R) = \( 14 \frac{2}{7} \% = \frac{100}{7} \% \)

### Step 2: Use the formula for Compound Interest
The formula for the amount (A) after time (T) with principal (P) and rate (R) is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Substituting the known values for the 13th year:
\[ 1024 = P \left(1 + \frac{100}{7 \times 100}\right)^{13} \]
This simplifies to:
\[ 1024 = P \left(1 + \frac{1}{7}\right)^{13} = P \left(\frac{8}{7}\right)^{13} \]

### Step 3: Express P in terms of 1024
Rearranging the equation gives:
\[ P = 1024 \cdot \left(\frac{7}{8}\right)^{13} \]

### Step 4: Calculate the amount at the end of the 10th year
Now, we need to find the compound interest for the 10th year. The amount at the end of 10 years (A10) can be expressed as:
\[ A_{10} = P \left(\frac{8}{7}\right)^{10} \]
Substituting P from the previous step:
\[ A_{10} = 1024 \cdot \left(\frac{7}{8}\right)^{13} \cdot \left(\frac{8}{7}\right)^{10} \]

### Step 5: Simplify the expression
This simplifies to:
\[ A_{10} = 1024 \cdot \left(\frac{7}{8}\right)^{3} \]
Calculating \(\left(\frac{7}{8}\right)^{3}\):
\[ \left(\frac{7}{8}\right)^{3} = \frac{343}{512} \]
Thus,
\[ A_{10} = 1024 \cdot \frac{343}{512} \]

### Step 6: Calculate A10
Calculating \(A_{10}\):
\[ A_{10} = 1024 \div 512 \cdot 343 = 2 \cdot 343 = 686 \]

### Step 7: Calculate Compound Interest for the 10th Year
The compound interest for the 10th year (CI10) is given by:
\[ CI_{10} = A_{10} - A_{9} \]
Where \(A_{9} = P \left(\frac{8}{7}\right)^{9}\).

### Step 8: Find A9
Using the same method:
\[ A_{9} = 1024 \cdot \left(\frac{7}{8}\right)^{4} = 1024 \cdot \frac{2401}{4096} \]
Calculating \(A_{9}\):
\[ A_{9} = 1024 \div 4096 \cdot 2401 = \frac{2401}{4} = 600.25 \]

### Step 9: Calculate CI10
Now we can find \(CI_{10}\):
\[ CI_{10} = 686 - 600.25 = 85.75 \]

### Final Answer
The compound interest for the 10th year is approximately **85.75**.

---