To solve the problem step by step, we will use the concept of compound interest.
### Step 1: Understand the given information
We know that Rs 60,000 grows to Rs 63,654 in a certain even number of years, compounded annually.
### Step 2: Set up the compound interest formula
The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
Where:
- \( A \) = the amount of money accumulated after n years, including interest.
- \( P \) = the principal amount (the initial amount of money).
- \( r \) = the annual interest rate (in percentage).
- \( t \) = the number of years the money is invested or borrowed.
### Step 3: Substitute the known values into the formula
Here, \( A = 63,654 \) and \( P = 60,000 \). We can set up the equation:
\[ 63,654 = 60,000 \left(1 + \frac{r}{100}\right)^t \]
### Step 4: Solve for \( \left(1 + \frac{r}{100}\right)^t \)
To isolate \( \left(1 + \frac{r}{100}\right)^t \), we divide both sides by 60,000:
\[ \left(1 + \frac{r}{100}\right)^t = \frac{63,654}{60,000} \]
Calculating the right side:
\[ \left(1 + \frac{r}{100}\right)^t = 1.0609 \]
### Step 5: Take the square root for half the period
Since we are interested in the amount after half the period, we will take the square root of both sides:
\[ 1 + \frac{r}{100} = \sqrt{1.0609} \]
Calculating the square root:
\[ 1 + \frac{r}{100} \approx 1.030 \]
### Step 6: Solve for \( r \)
Now, we can isolate \( r \):
\[ \frac{r}{100} \approx 1.030 - 1 \]
\[ \frac{r}{100} \approx 0.030 \]
\[ r \approx 3 \]
### Step 7: Calculate the amount for half the period
Now we will use the same formula to find the amount after half the period:
\[ A_{half} = 60,000 \left(1 + \frac{r}{100}\right)^{t/2} \]
Substituting \( 1 + \frac{r}{100} \) from Step 5:
\[ A_{half} = 60,000 \left(1.030\right)^{t/2} \]
Since \( \left(1.030\right)^{t/2} \) is the square root of \( 1.0609 \):
\[ A_{half} = 60,000 \times \sqrt{1.0609} \]
Calculating:
\[ A_{half} = 60,000 \times 1.030 \]
\[ A_{half} = 61,800 \]
### Final Answer
The amount would grow to Rs 61,800 if it is invested at the same rate for half the period.
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