To find out how many years it will take for an investment to grow from Birr 12,000 to Birr 24,000 with an annual interest rate of 7% compounded annually, we can use the formula for compound interest:
\[ A = P (1 + r)^t \]
Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (decimal).
- \( t \) is the number of years the money is invested or borrowed.
In this case:
- \( A = 24000 \)
- \( P = 12000 \)
- \( r = 0.07 \)
We need to solve for \( t \):
\[ 24000 = 12000 (1 + 0.07)^t \]
Dividing both sides by 12000 gives:
\[ 2 = (1.07)^t \]
Now we can take the logarithm of both sides to solve for \( t \):
\[ \log(2) = \log((1.07)^t) \]
Using the power rule of logarithms:
\[ \log(2) = t \cdot \log(1.07) \]
Now, solve for \( t \):
\[ t = \frac{\log(2)}{\log(1.07)} \]
Calculating this using logarithm values:
- \( \log(2) \approx 0.3010 \)
- \( \log(1.07) \approx 0.0291 \)
Now, substituting these values into the equation:
\[ t \approx \frac{0.3010}{0.0291} \approx 10.34 \]
Thus, it will take approximately 10.34 years for Birr 12,000 to grow to Birr 24,000 at an annual interest rate of 7% compounded annually.