I'm stuck at trying to understand the answer to this problem related with inflation. Can someone enlighten me with the proper interpretation of it?.
$\text{The problem is as follows:}$ $\text{In a certain country located in Asia, the inflation in September was 10% and the inflation}$ $\text{in October is 5%. What is the accumulated inflation during these two months?}$
Common sense (I believe) would dictate to sum both like this:
$\textrm{Accumulated inflation}=\textrm{Inflation in Septemeber}+\textrm{Inflation in October}$
Therefore,
$\textrm{Accumulated inflation}=10\%+5\%=15\% $
However by checking the answers from my book tells me I'm wrong since the correct answer is $15.5\%$ and not $15\%$. Which part is not correct in my interpretation?.
If I have goods worth $1,000D$ at the beginning of September, they are worth $1,100D$ at the end of September, and October inflation applies to this figure, so that at the end of October, they are worth $1,155D$.
The calculation is $(1+0.1)\times (1+0.05)=1.155=1+0.155$
So if the cost of goods at the beginning of September is $C$ and the inflation rate for September is $i_S \%$, then the cost of those same goods at the end of September is $$C+\frac {i_S}{100}C=\left(1+\frac {i_S}{100}\right)C$$
Now the cost of goods at the beginning of October is $\left(1+\frac {i_S}{100}\right)C$ and if inflation duding October is $i_O\%$ the same logic applies and the price at the end of October is $$\left(1+\frac {i_O}{100}\right)\left(1+\frac {i_S}{100}\right)C$$ and this is $$\left(1+\frac {i_S+i_O}{100}+\frac {i_S\cdot i_O}{10000}\right)C$$
The second term here involves adding the interest rates, which is what you would do if you were applying simple interest (inflation). But here we have compound interest (inflation). The price at the end of September is higher than the price at the beginning of the month, and the inflation in October applies not just to the price at the beginning of September, but also to the increase due to September inflation. So there is an adjustment to be made.
Think what happens if prices double in September and then double again in October, if you want a simple example.