What is the compound interest if the money put in the bank, with the required compound interest in $a$ years increases by $b%$

Question

The task is to write a program which for given $a$ and $b$ returns what is the compound interest if the money put in the bank with the required compound interest in $a$ years increases by $b\%$.

I do not see how can I find the compound interest if I do not know the starting amount of money?

Could it be $(1+\frac{b}{100})^{a}$?

Answer 1

Increasing by $b\%$ means:

$$b=\frac{\text{Final Amount – Initial amount}}{\text{Initial amount}} \cdot 100\% \tag{1}$$

Now let final amount be $M_f$ and intial amount be $M_i$. substituiting this in $(1)$, we rewrite as

$$b=\frac{M_f-M_i}{M_i} \cdot 100\% $$ $$\implies \frac{M_i(100+b)}{100}= M_f \tag{2}$$ Now for a compound interest $r\%$ per year, for $a$ years, we get $M_f$ as:

$$M_f=M_i\left(1+\frac{r}{100}\right)^a$$

Substituting this in $(2)$, we get

$$\left(1+\frac{b}{100}\right)=\left(1+\frac{r}{100}\right)^a$$

$$\therefore r=100\left(\left(1+\frac{b}{100}\right)^{1/a}-1\right)$$

However you are correct in the fact you can't find the value of $M_f$ if you do not know $M_i$. You get $r$ only.

Answer 2

You are correct in your presumption. Compound Interest depends on the principal and thus you will need it to find the compound interest as the formula dictates $$\text{Amount} = P\left ( 1 + \frac {\text{rate}}{100} \right )^a$$ $$\implies \text{C.I} = P\left\{\left(1 + \frac {\text{rate}}{100}\right)^a - 1\right\}$$