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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?.

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3 Answers

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Assume you are interested in an item that costs

1) 100 USD at the beginning of September.

At the end of September the same item is sold for

2) 110 USD , I.e 10% inflation.

At the end of October this item is sold for

110 + (5/100)110 USD = 110(1.05)USD=

(105 +10.5) USD = 115.5 USD, I.e 5% inflation in October.

"Combined " inflation:

Total price increase; 115.5 -100 USD =15.5 USD.

Original price: 100 USD

Total price increase in % for the combined period?

Can you do it?

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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 $(1+0.1)\times (1+0.05)=1.155=1+0.155$

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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.

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You can do it by summing like this :

1/ suppose our initial price is : B

2/First mounth inflation = 10% of B (at the end of 1st mounth: new price B'=1.1 B= 110% B)

3/Second mounth inflation= 5% of B'(the initial price at mouth 2 is the new price B') = 5% * 110% B = 5.5% of B

4/ Cumulative inflation over 2 mounth = 10% of B + 5.5% of B = 15.5% of B.

So inflation of a certain mounth will be applied to all ready inflated price from other mounths and not to the initial price of the first mounth. In fact only the first mounth (or year or any term) will be simply applied, that's the conpounding effect. (so to your calculation you can't just sum 10% of B and 5% of B' but rather use 5% of B' = 5.5% of B)

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