Why do we need to divide by the CPI of year of reference when calculating inflation rate?

Question

I know that inflation rate = $$ CPI_{n} - CPI_{n-1} \over CPI_{n-1}$$

but I wonder what's the point of dividing by $CPI_{n-1}$, CPI is already a percentage so inflation rate should be $ CPI_{n} - CPI_{n-1}$ directly (eg. Assume base year is 2003 and CPI in 2010 was 105% and CPI in 2021 was 120% then should not inflation rate be 120% - 105% = 15%?)

I suspect we do that because we often tend to think of CPI as in dollars instead of percentage, so for example we can say that what could be bought with 1 dollar in 2003 would be bought with 1.20 dollars in 2021 so we still need to divide by $CPI_{n-1}$ when calculating inflation rate? but I'm still confused tho.

Thanks in advance (and please ignore how unrealistic my example was, I made that up for demonstration)

Answer 1

Because inflation is defined as a positive growth rate of CPI.

By definition growth rate is:

$$g=\frac{x_t-x_{t-1}}{x_{t-1}} \tag{*}$$

That is just how growth rate is defined in sciences anywhere from physics or biology to social sciences.

It does not matter if the number is in percentages. A growth rate of interest rate is still calculated by the same equation *.

$x_t-x_{t-1}$ is just simple change, not growth rate.

Assume base year is 2003 and CPI in 2010 was 105% and CPI in 2021 was 120% then should not inflation rate be 120% - 105% = 15%?

No in that problem CPI changed by 15 points, inflation rate in your example is $0.1429$ or $14.29\%$. Note inflation does not need to be expressed in per cent terms - in many research papers it’s just in decimals.