A sum of 9, 600 is invested for 3 years at 10%
p.a. compound interest.
(i) What is the sum due at the end of the first year?
(ii) What is the sum due at the end of the second
year?
(iii) Find the compound interest earned in the first
2 years.



Answer :

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.