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Sagot :
Certainly, let's determine how much Khadija has in her account after 6 years given an initial investment of R25,000 with an annual interest rate of 10.3%, compounded quarterly.
### Given:
- Principal (P): R25,000
- Annual interest rate (r): 10.3% or 0.103
- Compounding frequency (n): Quarterly, which means the interest is compounded 4 times a year.
- Time (t): 6 years
### Step-by-step solution:
1. Understanding Compound Interest Formula:
The formula for compound interest is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Where:
- [tex]\( A \)[/tex] is the amount of money accumulated after the specified time, including interest.
- [tex]\( P \)[/tex] is the principal amount (the initial amount of money).
- [tex]\( r \)[/tex] is the annual interest rate (decimal).
- [tex]\( n \)[/tex] is the number of times the interest is compounded per year.
- [tex]\( t \)[/tex] is the time the money is invested for, in years.
2. Substitute the given values into the formula:
- Principal (P): R25,000
- Annual interest rate (r): 0.103 (10.3% as a decimal)
- Compounding frequency (n): 4 (compounded quarterly)
- Time (t): 6 years
So, we substitute these values into the formula:
[tex]\[ A = 25000 \left(1 + \frac{0.103}{4}\right)^{4 \times 6} \][/tex]
3. Calculate each part of the formula:
- Rate per period: [tex]\(\frac{0.103}{4}\)[/tex]
- This simplifies to [tex]\(\frac{0.103}{4} = 0.02575\)[/tex]
- Number of compounding periods: [tex]\(4 \times 6\)[/tex]
- This simplifies to [tex]\(4 \times 6 = 24\)[/tex]
- Base of the exponential part: [tex]\(1 + 0.02575\)[/tex]
- This simplifies to [tex]\(1 + 0.02575 = 1.02575\)[/tex]
- Exponent: [tex]\(24\)[/tex]
4. Applying the exponent:
[tex]\[ A = 25000 \left(1.02575\right)^{24} \][/tex]
5. Calculating the final amount:
- By evaluating this expression, we find:
[tex]\[ A \approx 46018.94 \][/tex]
Therefore, after 6 years, Khadija has approximately R46,018.94 in her account.
### Given:
- Principal (P): R25,000
- Annual interest rate (r): 10.3% or 0.103
- Compounding frequency (n): Quarterly, which means the interest is compounded 4 times a year.
- Time (t): 6 years
### Step-by-step solution:
1. Understanding Compound Interest Formula:
The formula for compound interest is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Where:
- [tex]\( A \)[/tex] is the amount of money accumulated after the specified time, including interest.
- [tex]\( P \)[/tex] is the principal amount (the initial amount of money).
- [tex]\( r \)[/tex] is the annual interest rate (decimal).
- [tex]\( n \)[/tex] is the number of times the interest is compounded per year.
- [tex]\( t \)[/tex] is the time the money is invested for, in years.
2. Substitute the given values into the formula:
- Principal (P): R25,000
- Annual interest rate (r): 0.103 (10.3% as a decimal)
- Compounding frequency (n): 4 (compounded quarterly)
- Time (t): 6 years
So, we substitute these values into the formula:
[tex]\[ A = 25000 \left(1 + \frac{0.103}{4}\right)^{4 \times 6} \][/tex]
3. Calculate each part of the formula:
- Rate per period: [tex]\(\frac{0.103}{4}\)[/tex]
- This simplifies to [tex]\(\frac{0.103}{4} = 0.02575\)[/tex]
- Number of compounding periods: [tex]\(4 \times 6\)[/tex]
- This simplifies to [tex]\(4 \times 6 = 24\)[/tex]
- Base of the exponential part: [tex]\(1 + 0.02575\)[/tex]
- This simplifies to [tex]\(1 + 0.02575 = 1.02575\)[/tex]
- Exponent: [tex]\(24\)[/tex]
4. Applying the exponent:
[tex]\[ A = 25000 \left(1.02575\right)^{24} \][/tex]
5. Calculating the final amount:
- By evaluating this expression, we find:
[tex]\[ A \approx 46018.94 \][/tex]
Therefore, after 6 years, Khadija has approximately R46,018.94 in her account.
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