The annual rate of interest is 4.80%
Step-by-step explanation:
The formula for compound interest, including principal sum is
[tex]A=P(1+\frac{r}{n})^{nt}[/tex] where:
Suppose that time invest $10,000 in an account that offers are percent annual interest, compounded quarterly if the investment increases to $12,694.34 in five years
∵ P = $10,000
∵ A = $12,694.34
∵ n = 4 ⇒ compounded quarterly
∵ t = 5 years
- Substitute all these values in the formula above
∴ [tex]12,694.34=10,000(1+\frac{r}{4})^{4(5)}[/tex]
∴ [tex]12,694.34=10,000(1+\frac{r}{4})^{20}[/tex]
- Divide both sides by 10,000
∴ [tex]1.269434=(1+\frac{r}{4})^{20}[/tex]
- Insert ㏒ to both sides
∴ [tex]log(1.269434)=log(1+\frac{4}{n})^{20}[/tex]
∴ [tex]log(1.269434)=20log(1+\frac{4}{n})[/tex]
- Divide both sides by 20
∴ [tex]0.00518=log(1+\frac{4}{n})[/tex]
- Remember [tex]log_{a}b=c[/tex] can be written as [tex]a^{c}=b[/tex]
∵ The base of the ㏒ is 10
∴ [tex]10^{0.00518}=(1+\frac{r}{4})[/tex]
∴ [tex]1.011998806=1+\frac{r}{4}[/tex]
- Subtract 1 from both sides
∴ [tex]0.011998806=\frac{r}{4}[/tex]
- Multiply both sides by 4
∴ 0.04799522 = r
∵ r is the rate in decimal
- To find the annual rate of interest R% multiply r by 100%
∴ R% = 0.04799522 × 100% = 4.799522%
∴ R% ≅ 4.80%
The annual rate of interest is 4.80%
Learn more:
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