Question

A certain company recently sold five-year $1000 bonds with an annual yield of 9.75%. After how much time could they be sold for twice their original price? Give your answer in years and months. (Round your answer to the nearest month.

Answer 1

Answer:

After 7 years and 5 months.

Step-by-step explanation:

Let x represent number of years.

We have been given that a certain company recently sold five-year $1000 bonds with an annual yield of 9.75%.

We can see that the value of bond is increasing exponentially, so we will use exponential growth formula to solve our given problem.

$y=a\cdot (1+r)^x$, where,

y = Final value,

a = Initial value,

r = Rate in decimal form,

x = Time

$9.75\%=\frac{9.75}{100}=0.0975$

Substituting given values:

$y=1000\cdot (1+0.0975)^x$

$y=1000\cdot (1.0975)^x$

Since we need the selling price to be twice the original price, so we will substitute $y=2000$ in above equation as:

$2000=1000\cdot (1.0975)^x$

$\frac{2000}{1000}=\frac{1000\cdot (1.0975)^x}{1000}$

$2=1.0975^x$

Switch sides:

$1.0975^x=2$

Take natural log of both sides:

$\text{ln}(1.0975^x)=\text{ln}(2)$

Applying rule $\text{ln}(a^b)=b\cdot \text{ln}(a)$:

$x\cdot \text{ln}(1.0975)=\text{ln}(2)$

$\frac{x\cdot \text{ln}(1.0975)}{\text{ln}(1.0975)}=\frac{\text{ln}(2)}{\text{ln}(1.0975)}$

$x=\frac{0.6931471805599453}{0.0930348659671894}$

$x=7.45040231265$

$x\approx 7.4504$

Since x represents time in years, so we need to convert decimal part into months by multiplying .4504 by 12 as 1 year equals 12 months.

7 years and 12*0.4504023 months = 7 years 5.4 months = 7 years 5 months

Therefore, after 7 years and 5 months the company could sold the bonds for twice their original price.