# 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!

Homework Helper
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)

Homework Helper
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?

Last edited:
mcbates
how does that end up being 2.3 years

Homework Helper
mcbates said:
how does that end up being 2.3 years
$$n = \frac{ln(\frac{5000}{4000})}{ln(1,0975)}$$

Homework Helper
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?

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

mcbates
okayyyy! Thanks! :rofl:

Homework Helper
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

Homework Helper
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

Homework Helper
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.