To solve this problem, we need to find the initial principal amount that grows to the given final amounts under the specified conditions of compound interest. We will use the formula for compound interest, which is:
[tex]\[ A = P(1 + r)^t \][/tex]
where:
- [tex]\( A \)[/tex] is the final amount
- [tex]\( P \)[/tex] is the principal amount (the initial sum of money)
- [tex]\( r \)[/tex] is the annual interest rate (as a decimal)
- [tex]\( t \)[/tex] is the time the money is invested for (in years)
### Part (i):
Final Amount: [tex]\( A = 1323 \)[/tex]
Time Period: [tex]\( t = 2 \)[/tex] years
Annual Interest Rate: [tex]\( r = 5\% = 0.05 \)[/tex]
We need to find the principal amount [tex]\( P \)[/tex] such that:
[tex]\[ 1323 = P(1 + 0.05)^2 \][/tex]
First, calculate the growth factor:
[tex]\[ (1 + 0.05)^2 = 1.05^2 = 1.1025 \][/tex]
Next, isolate [tex]\( P \)[/tex]:
[tex]\[ P = \frac{1323}{1.1025} = 1200 \][/tex]
So, the principal amount that becomes Rs 1,323 in 2 years at a compound interest rate of 5% is Rs 1200.
### Part (ii):
Final Amount: [tex]\( A = 6655 \)[/tex]
Time Period: [tex]\( t = 3 \)[/tex] years
Annual Interest Rate: [tex]\( r = 10\% = 0.1 \)[/tex]
We need to find the principal amount [tex]\( P \)[/tex] such that:
[tex]\[ 6655 = P(1 + 0.1)^3 \][/tex]
First, calculate the growth factor:
[tex]\[ (1 + 0.1)^3 = 1.1^3 \approx 1.331 \][/tex]
Next, isolate [tex]\( P \)[/tex]:
[tex]\[ P = \frac{6655}{1.331} \approx 4999.999999999998 \][/tex]
So, the principal amount that becomes Rs 6,655 in 3 years at a compound interest rate of 10% is approximately Rs 5000.
### Conclusion:
The sum of money which becomes:
(i) Rs 1,323 in 2 years at a compound interest rate of 5% is Rs 1200.
(ii) Rs 6,655 in 3 years at a compound interest rate of 10% is approximately Rs 5000.
