Sure! Let's approach this step by step using the compound interest formula:
The formula for compound interest is:
[tex]\[ A = P (1 + \frac{r}{n})^{nt} \][/tex]
where:
- [tex]\( P \)[/tex] is the principal amount (9600 in this case)
- [tex]\( r \)[/tex] is the annual interest rate (expressed as a decimal, so 10% becomes 0.10)
- [tex]\( n \)[/tex] is the number of times the interest is compounded per year (since it's compounded annually in this case, [tex]\( n = 1 \)[/tex])
- [tex]\( t \)[/tex] is the number of years the money is invested (3 years)
However, since we're calculating per annum (annually), we can simplify the formula to:
[tex]\[ A = P (1 + r)^t \][/tex]
Let's calculate each part of the question:
### (i) The sum due at the end of the first year:
Given:
- [tex]\( P = 9600 \)[/tex]
- [tex]\( r = 0.10 \)[/tex]
- [tex]\( t = 1 \)[/tex] (since we are interested in the first year)
Using the formula:
[tex]\[ A_1 = 9600 \times (1 + 0.10)^1 \][/tex]
[tex]\[ A_1 = 9600 \times 1.10 \][/tex]
[tex]\[ A_1 = 10560 \][/tex]
So, the sum due at the end of the first year is ₦10,560.
### (ii) The sum due at the end of the second year:
Given:
- [tex]\( P = 9600 \)[/tex]
- [tex]\( r = 0.10 \)[/tex]
- [tex]\( t = 2 \)[/tex] (for the end of the second year)
Using the formula:
[tex]\[ A_2 = 9600 \times (1 + 0.10)^2 \][/tex]
[tex]\[ A_2 = 9600 \times (1.10)^2 \][/tex]
[tex]\[ A_2 = 9600 \times 1.21 \][/tex]
[tex]\[ A_2 = 11616 \][/tex]
So, the sum due at the end of the second year is ₦11,616.
### (iii) The compound interest earned in the first 2 years:
Compound interest earned is the amount accumulated minus the principal.
For the end of two years:
- The amount accumulated at the end of two years is ₦11,616.
- The principal is ₦9600.
Thus, the compound interest earned in the first two years is:
[tex]\[ \text{Compound Interest} = A_2 - P \][/tex]
[tex]\[ \text{Compound Interest} = 11616 - 9600 \][/tex]
[tex]\[ \text{Compound Interest} = 2016 \][/tex]
So, the compound interest earned in the first 2 years is ₦2,016.
In summary:
1. The sum due at the end of the first year is ₦10,560.
2. The sum due at the end of the second year is ₦11,616.
3. The compound interest earned in the first 2 years is ₦2,016.
