In how many years will ₹ 25,000 yield ₹ 8,275 as compound interest at 10% per annum compounded annually?

Options

A. 2
B. 3
C. 5
D. 4

Answer

To solve the problem of how many years ₹ 25,000 will yield ₹ 8,275 as compound interest at 10% per annum compounded annually, we can follow these steps:

### Step 1: Identify the given values
- Principal (P) = ₹ 25,000
- Compound Interest (CI) = ₹ 8,275
- Rate of interest (R) = 10% per annum

### Step 2: Calculate the total amount (A)
The total amount (A) after the interest is added can be calculated using the formula:
\[ A = P + CI \]
Substituting the values:
\[ A = 25000 + 8275 = 33275 \]

### Step 3: Set up the ratio of Principal to Amount
We need to find the ratio of the Principal (P) to the Amount (A):
\[ \text{Ratio} = \frac{P}{A} = \frac{25000}{33275} \]

### Step 4: Simplify the ratio
To simplify the ratio, we can divide both the numerator and denominator by 25:
\[ \frac{25000}{33275} = \frac{1000}{1331} \]

### Step 5: Relate the ratio to the compound interest formula
The relationship between Principal and Amount in terms of time (t) can be expressed as:
\[ \frac{P}{A} = \left(1 + \frac{R}{100}\right)^{-t} \]
Where \( R \) is the rate of interest. In this case, \( R = 10\% \), so:
\[ \frac{P}{A} = \left(1 + \frac{10}{100}\right)^{-t} = \left(1.1\right)^{-t} \]

### Step 6: Set up the equation
From our earlier ratio, we have:
\[ \frac{1000}{1331} = \left(1.1\right)^{-t} \]

### Step 7: Convert the ratio to exponential form
We can express the ratio as:
\[ 1000 = 10^3 \quad \text{and} \quad 1331 = 11^3 \]
Thus, we can write:
\[ \left(\frac{10}{11}\right)^3 = \left(1.1\right)^{-t} \]

### Step 8: Equate the exponents
Since both sides have the same base, we can equate the exponents:
\[ 3 = -t \cdot \log(1.1) \]
To find \( t \):
\[ t = 3 \]

### Conclusion
The time required for ₹ 25,000 to yield ₹ 8,275 as compound interest at 10% per annum compounded annually is **3 years**.

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