To answer this question, we need to use the formula for compound interest. The formula for the future value (FV) of an investment is:
[tex]\[ FV = P \left(1 + \frac{r}{n}\right)^{n \cdot t} \][/tex]
where:
- [tex]\( FV \)[/tex] is the future value of the investment (Br. 10,000).
- [tex]\( P \)[/tex] is the principal amount (the initial deposit, which we need to find).
- [tex]\( r \)[/tex] is the annual interest rate (10% or 0.10).
- [tex]\( n \)[/tex] is the number of compounding periods per year (4, since it is compounded quarterly).
- [tex]\( t \)[/tex] is the time in years (10 years).
We need to solve for [tex]\( P \)[/tex], the principal amount, using the provided values.
First, calculate the quarterly interest rate:
[tex]\[ \frac{r}{n} = \frac{0.10}{4} = 0.025 \][/tex]
Next, determine the total number of compounding periods:
[tex]\[ n \cdot t = 4 \cdot 10 = 40 \][/tex]
Now, rearrange the compound interest formula to solve for [tex]\( P \)[/tex]:
[tex]\[ P = \frac{FV}{\left(1 + \frac{r}{n}\right)^{n \cdot t}} \][/tex]
Substituting the known values:
[tex]\[ P = \frac{10,000}{\left(1 + 0.025\right)^{40}} = \frac{10,000}{\left(1.025\right)^{40}} \][/tex]
Calculate [tex]\( (1.025)^{40} \)[/tex]:
[tex]\[ (1.025)^{40} \approx 2.685 \][/tex]
Therefore, the principal amount [tex]\( P \)[/tex]:
[tex]\[ P = \frac{10,000}{2.685} \approx 3,725.41 \][/tex]
Thus, the initial deposit should be approximately Br. 3,725.41.
Next, to find the amount of compound interest earned after 10 years, we can use the future value and the principal amount:
[tex]\[ \text{Compound Interest} = FV - P \][/tex]
Substitute the known values:
[tex]\[ \text{Compound Interest} = 10,000 - 3,725.41 = 6,274.59 \][/tex]
So, the amount of compound interest earned after 10 years is Br. 6,274.59.
To summarize:
- The initial deposit required to have a balance of Br. 10,000 after 10 years with a 10% annual interest rate compounded quarterly is approximately Br. 3,725.41.
- The amount of compound interest earned after 10 years is Br. 6,274.59.
