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Sagot :
Let's solve the problem step-by-step to find the nominal annual rate of interest at which money will double itself in six years and six months, given it is compounded semi-annually.
### Step-by-Step Solution:
1. Identify the Variables:
- Principal (P): The initial amount of money invested or loaned, which we can assume to be 1 unit of currency for ease.
- Final Amount (A): The amount of money that the investment or loan grows to, which in this case is double the principal. Thus, [tex]\( A = 2 \times P \)[/tex].
- Time (t): The time period for which the money is invested, which is six years and six months, or 6.5 years.
- Compounding Frequency (n): The number of times interest is compounded per year. Since it is compounded semi-annually, [tex]\( n = 2 \)[/tex].
2. Compound Interest Formula:
The formula to calculate the amount [tex]\( A \)[/tex] after time [tex]\( t \)[/tex] with a principal [tex]\( P \)[/tex], at an annual nominal interest rate [tex]\( r \)[/tex], compounded [tex]\( n \)[/tex] times per year is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
3. Plug in the Known Values:
- [tex]\( A = 2P \)[/tex]
- [tex]\( P = 1 \)[/tex]
- [tex]\( t = 6.5 \)[/tex]
- [tex]\( n = 2 \)[/tex]
Now the equation looks like this:
[tex]\[ 2 = \left(1 + \frac{r}{2}\right)^{2 \times 6.5} \][/tex]
[tex]\[ 2 = \left(1 + \frac{r}{2}\right)^{13} \][/tex]
4. Solve for [tex]\( r \)[/tex]:
To isolate [tex]\( r \)[/tex], take the 13th root of both sides to get rid of the exponent:
[tex]\[ \left(1 + \frac{r}{2}\right) = 2^{\frac{1}{13}} \][/tex]
[tex]\[ 1 + \frac{r}{2} \approx 1.0547660764816467 \][/tex]
5. Isolate [tex]\( r \)[/tex]:
Subtract 1 from both sides:
[tex]\[ \frac{r}{2} \approx 1.0547660764816467 - 1 \][/tex]
[tex]\[ \frac{r}{2} \approx 0.0547660764816467 \][/tex]
Multiply both sides by 2:
[tex]\[ r \approx 2 \times 0.0547660764816467 \][/tex]
[tex]\[ r \approx 0.10953215296329333 \][/tex]
Therefore, the nominal annual rate of interest needed for the investment to double itself in six years and six months, if compounded semi-annually, is approximately 10.95%.
### Step-by-Step Solution:
1. Identify the Variables:
- Principal (P): The initial amount of money invested or loaned, which we can assume to be 1 unit of currency for ease.
- Final Amount (A): The amount of money that the investment or loan grows to, which in this case is double the principal. Thus, [tex]\( A = 2 \times P \)[/tex].
- Time (t): The time period for which the money is invested, which is six years and six months, or 6.5 years.
- Compounding Frequency (n): The number of times interest is compounded per year. Since it is compounded semi-annually, [tex]\( n = 2 \)[/tex].
2. Compound Interest Formula:
The formula to calculate the amount [tex]\( A \)[/tex] after time [tex]\( t \)[/tex] with a principal [tex]\( P \)[/tex], at an annual nominal interest rate [tex]\( r \)[/tex], compounded [tex]\( n \)[/tex] times per year is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
3. Plug in the Known Values:
- [tex]\( A = 2P \)[/tex]
- [tex]\( P = 1 \)[/tex]
- [tex]\( t = 6.5 \)[/tex]
- [tex]\( n = 2 \)[/tex]
Now the equation looks like this:
[tex]\[ 2 = \left(1 + \frac{r}{2}\right)^{2 \times 6.5} \][/tex]
[tex]\[ 2 = \left(1 + \frac{r}{2}\right)^{13} \][/tex]
4. Solve for [tex]\( r \)[/tex]:
To isolate [tex]\( r \)[/tex], take the 13th root of both sides to get rid of the exponent:
[tex]\[ \left(1 + \frac{r}{2}\right) = 2^{\frac{1}{13}} \][/tex]
[tex]\[ 1 + \frac{r}{2} \approx 1.0547660764816467 \][/tex]
5. Isolate [tex]\( r \)[/tex]:
Subtract 1 from both sides:
[tex]\[ \frac{r}{2} \approx 1.0547660764816467 - 1 \][/tex]
[tex]\[ \frac{r}{2} \approx 0.0547660764816467 \][/tex]
Multiply both sides by 2:
[tex]\[ r \approx 2 \times 0.0547660764816467 \][/tex]
[tex]\[ r \approx 0.10953215296329333 \][/tex]
Therefore, the nominal annual rate of interest needed for the investment to double itself in six years and six months, if compounded semi-annually, is approximately 10.95%.
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