Discover a world of knowledge at Westonci.ca, where experts and enthusiasts come together to answer your questions. Connect with a community of experts ready to help you find accurate solutions to your questions quickly and efficiently. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
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.
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.