How to solve the bank interest as monthly

Question

I am trying to solve the following:

Suppose the bank paid 12 % per year, but compounded that interest monthly. That is, suppose 1 % interest was added to your account every month. Then how much would you have after 30 years and after 60 years if you started with $100?

What I did is uisng this formula:

$$ y[n]=(1.001^{12})*y[0] $$

where $ y[0] = 100\$ $

The answer for 30 years was $~143\$ $

but it is wrong as I've been said.

When I compound your interest monthly this means that I give you a 1/12th of the total years rate every single month. In this specific example we have a 12% interest rate so that would mean that every single month your account is going to grow by 1%. Now, we are doing this 12 times a year for 30 years

Still complicating for me.

Answer 1

Note that monthly interest $1$% is not equivalent to that of yearly $12$% because$$\text{interests per month after 1 year}=(1.01)^{12}\approx1.1268\\\text{interests per year after 1 year}=(1.12)^{1}=1.12$$so we can see$$\text{1% interest per month in one year}≠\text{12% interest per annum}$$therefore if our interest is $1$% per month our deposit after $30$ years and $60$ years starting with $100\$$ would be$$M_{30}=(1.01)^{30\times 12}\approx3594.96\$\\M_{60}=(1.01)^{30\times 12}\approx129237.67\$$$and by yearly $12$% interest we have$$M_{30}=(1.12)^{30}\approx2995.99\$\\M_{60}=(1.12)^{60}\approx89759.69\$$$

Answer 2

Remember that the compound interest formula is $C = P\left(1+\dfrac{r}{n}\right)^{nt}$ where $r$ is the rate, $n$ is the amount of times during the year it's compounded, $t$ is the number of years, and $P$ is the principle, or initial starting value.

Your rate $r$ is $0.12$, or $12%$. When you divide that by the $12$ months you're compounding interest, you should get $0.01$, not $0.001$.

Also notice the exponent, which should be $nt$. Since you're compounding $12$ times a year, and for $30$ years, you exponent should be $360$:

$$C = 100\left(1+\dfrac{0.12}{12}\right)^{12\cdot30} = 100(1.01)^{360}\approx3594.96$$