At the rate of 15% compound interest, the ratio of Amount and Principle is 529:400 in certain time period then find the time period?

Options

A. 2.5 year
B. 2 year
C. 1.5 year
D. 3 year

Answer

To solve the problem, we need to find the time period when the compound interest is applied at a rate of 15%, and the ratio of the amount to the principal is given as 529:400.

### Step-by-Step Solution:

1. **Understanding the Ratios**:
We know that the ratio of the amount (A) to the principal (P) is given as:
\[ \frac{A}{P} = \frac{529}{400} \]

2. **Using the Compound Interest Formula**:
The formula for the amount in compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
where \( r \) is the rate of interest and \( t \) is the time in years.

3. **Substituting the Values**:
Here, the rate \( r = 15\% \). We can express it as a fraction:
\[ \frac{r}{100} = \frac{15}{100} = \frac{3}{20} \]
Thus, the formula becomes:
\[ A = P \left(1 + \frac{3}{20}\right)^t \]
Simplifying further:
\[ A = P \left(\frac{23}{20}\right)^t \]

4. **Setting Up the Equation**:
From the ratio given:
\[ \frac{A}{P} = \frac{23^t}{20^t} = \frac{529}{400} \]
This implies:
\[ \left(\frac{23}{20}\right)^t = \frac{529}{400} \]

5. **Identifying Powers**:
We can express \( 529 \) and \( 400 \) in terms of their bases:
\[ 529 = 23^2 \quad \text{and} \quad 400 = 20^2 \]
Therefore, we can rewrite the equation as:
\[ \left(\frac{23}{20}\right)^t = \left(\frac{23}{20}\right)^2 \]

6. **Equating the Exponents**:
Since the bases are the same, we can equate the exponents:
\[ t = 2 \]

7. **Conclusion**:
The time period is \( t = 2 \) years.

### Final Answer:
The time period is **2 years**.