Find the compound interest for third year on certain amount at the rate of 8% if CI for second year is 8960?

Options

A. ₹ 9676.80
B. ₹9670.80
C. ₹ 9766.80
D. ₹9760.80

Answer

To find the compound interest for the third year on a certain amount at the rate of 8%, given that the compound interest for the second year is 8,960, we can follow these steps:

### Step 1: Understand the relationship between the amounts
The compound interest for the second year (CI for Year 2) is given as 8,960. This means that the amount at the end of the second year (A1) is equal to the principal amount (P) multiplied by (1 + r/100)², where r is the rate of interest.

### Step 2: Set up the equation for the second year
We can express this as:
\[ A1 = P \times \left(1 + \frac{r}{100}\right)^2 \]
Given that r = 8%, we can substitute this into the equation:
\[ A1 = P \times \left(1 + \frac{8}{100}\right)^2 \]
\[ A1 = P \times \left(1.08\right)^2 \]

### Step 3: Set up the equation for the third year
The amount at the end of the third year (A2) can be expressed as:
\[ A2 = P \times \left(1 + \frac{r}{100}\right)^3 \]
Substituting r = 8% gives us:
\[ A2 = P \times \left(1.08\right)^3 \]

### Step 4: Relate A2 and A1
To find the relationship between A2 and A1, we can divide the two equations:
\[ \frac{A2}{A1} = \frac{P \times (1.08)^3}{P \times (1.08)^2} \]
The P cancels out:
\[ \frac{A2}{A1} = \frac{(1.08)^3}{(1.08)^2} = 1.08 \]

### Step 5: Substitute A1
From the previous steps, we know A1 = 8,960. Therefore:
\[ A2 = A1 \times 1.08 \]
\[ A2 = 8960 \times 1.08 \]

### Step 6: Calculate A2
Calculating the value:
\[ A2 = 8960 \times 1.08 = 9676.8 \]

### Step 7: Find the compound interest for the third year
The compound interest for the third year (CI for Year 3) is the amount at the end of the third year (A2) minus the amount at the end of the second year (A1):
\[ CI_{Year 3} = A2 - A1 \]
\[ CI_{Year 3} = 9676.8 - 8960 \]
\[ CI_{Year 3} = 716.8 \]

Thus, the compound interest for the third year is **716.8**.

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