Question

How long does it take for an investment to triple in value if it is invested at 10 % compounded quarterly?

Answer 1

Do you know how to calculate the compound interest?

Answer 2

A = P(1 + r/n)nt

Answer 3

Yes, that is correct.

Now what is A, r, n and t in that?

btw, in your question is the rate of interest 10% per annum (or per quarter)?

Answer 4

I think it might be per annum because it is asking for a number of years as the answer

Answer 5

Okay, I'll take the 10% to be per annum.

In the formula \(\displaystyle A = P\left ( 1 + \frac{r}{n} \right)^{nt}\),

\(A\) is the amount after \(t\) years,
\(r\) is the rate of interest per annum and
\(n\) is the number of times the interest is calculated in a year.
\(t\) is the number of years required.

Answer 6

r=0.10, n=4

Answer 7

I don't know what A or t would be

Answer 8

Yes

that's good for now.

Answer 9

Now, you want the total value to be three times of the invested amount. Here the invested amount is \(P\).

And the amount = total value = \(3 \times P\)

Make sense ?

Answer 10

Yeah that makes sense

Answer 11

great. Now, you want to find out how many years it will take to get that

Let's plug in the values now:
\(A = 3P\), \(r = 0.1\) and \(n = 4\) to get

\(\displaystyle 3\cancel{P} = \cancel{P}\left(1 + \frac{0.1}{4}\right)^{4t} \)
\(\displaystyle \left(1 + 0.025\right)^{4t} = 3\) or \( 1.025^{4t} = 3\)

everything fine till now?

Answer 12

Yes that looks good

Answer 13

Okay, now the major problem comes because that \(t\) is a power.

So, you can take the \(\log\) on both sides to make it easier!

\(\log(1.025)^{4t} = \log(3)\)

Now \( \log(a^b) = b \log (a)\), using this we can simplify the equation to

\(4t \times \log(1.025) = \log(3)\)

Can you solve the rest now? using a calculator of course ;)

Answer 14

I ended up with 11.12 years

Answer 15

I got .0429t=.4771 and divided .4771 by .0429

Answer 16

That seems to be right :)

Answer 17

Awesome, thank you!

Answer 18

You're welcome!