Interest rate calculation help

[mcbates]

i am having problems figuring out how to find the answer to this problem:

nancy wants to invest $4000 in saving certificates that bear an interest rate of 9.75% per yr, compound semiannually. how long a time period should she choose inorder to save an amt of $5000?

the answer is approximately 2.3 yrs. i am not sure how to get this answer. pleasssseee help!

 

[TD]

What have you tried?
Do you know how to set up the equation somewhat?

 

[mcbates]

i think you use the equation Ao(1+r/n)^nt: i think this one is it.
or Ao e^rt (growth)

 

[TD]

Well, we have 4000 at the start and it increased by 9.75 percent, which we can express by multiplying with 1.0975, this for each year. We also know what we want to end up with, so we get the equation: $$4000 \cdot 1.0975^n = 5000$$

Here, n is the number of years. Now, could you solve it?

 

[mcbates]

how does that end up being 2.3 years

 

[Päällikkö]

how does that end up being 2.3 years

$$n = \frac{ln(\frac{5000}{4000})}{ln(1,0975)}$$

 

[TD]

Indeed, and that's approximately 2.39 (so I'd say 2.4 when rounding...)
Since the interest comes semianually, to get (at least) the 5000 you have to wait 2.5 years.

 

[mcbates]

where did you get 1.0975 from?

 

[TD]

Didn't you read post #4? I already included the equation for you

 

[mcbates]

okayyyy! Thanks! :rofl:

 

[TD]

No problem

But since you didn't set it up yourself, I hope you do understand it?
If not, don't hesitate to ask for further details!

 

[mcbates]

the only reason i was wondering was because if you set it up in the equation i gave you it would be (1+9.75/2)^2T,,,i think

 

[TD]

Well I don't fully understand that one, where did you get the "2" for n?

 

[mcbates]

because the rate of interest in compounded semiannually

 

[TD]

Then, I think, the equation should be:

$$4000\left( {1 + \frac{{0.0975}} {2}} \right)^{2t} = 5000$$

That gives approx 2.34

I assume this is correct, because in my earlier equation we didn't use the fact that the interest was semianually.