Respuesta :
Answer:
7.3 years.
Step-by-step explanation:
We are asked to find the time taken for 12,500 to grow to 20000 at 6.5% annual interest compounded quarterly. We will use compound interest formula to solve our given problem.
[tex]A=P(1+\frac{r}{n})^{nt}[/tex], where,
A = Final amount after t years,
P = Principal amount,
r = Annual interest rate in decimal form,
n = Number of times interest is compounded per year,
t = Time in years.
[tex]6.5\%=\frac{6.5}{100}=0.065[/tex]
Upon substituting our given values in above formula, we will get:
[tex]20,000=12,500(1+\frac{0.065}{4})^{4\cdot t}[/tex]
[tex]20,000=12,500(1+0.01625)^{4\cdot t}[/tex]
[tex]20,000=12,500(1.01625)^{4\cdot t}[/tex]
[tex]12,500(1.01625)^{4\cdot t}=20,000[/tex]
[tex]\frac{12,500(1.01625)^{4\cdot t}}{12,500}=\frac{20,000}{12,500}[/tex]
[tex](1.01625)^{4\cdot t}=1.6[/tex]
Now we will take natural log of both sides of equation as:
[tex]\text{ln}((1.01625)^{4\cdot t})=\text{ln}(1.6)[/tex]
Using property [tex]\text{ln}(a^b)=b\cdot \text{ln}(a)[/tex], we will get:
[tex]4\cdot t\cdot \text{ln}(1.01625)=\text{ln}(1.6)[/tex]
[tex]\frac{4\cdot t\cdot \text{ln}(1.01625)}{4\cdot \text{ln}(1.01625)}=\frac{\text{ln}(1.6)}{4\cdot \text{ln}(1.01625)}[/tex]
[tex]t=\frac{0.470003629245}{4\cdot0.016119381879}[/tex]
[tex]t=\frac{0.470003629245}{0.064477527516}[/tex]
[tex]t=7.28941768[/tex]
Upon rounding to nearest tenth of year, we will get:
[tex]t\approx 7.3[/tex]
Therefore, it will take approximately 7.3 years for 12,500 to grow to 20000.
Answer:
5.9 year
Step-by-step explanation:
It takes 5.9 years to grow to 20000 at 6.5% annual interest compounded quartely!
