In this brief note I present the rationale behind the formula for computing a price index (e.g. the Consumer Price Index or CPI) as well as the formula for obtaining inflation-adjusted prices.
Briefly, a price index is a normalized number whose aim is representing the value of a basket of goods or services in a given region and time. Here, the key term is the word normalized, for the purposes of this note by normalized we mean a quantity that is independent of a currency.
The textbook formula for the level of an index at time $t$ is given by
$$ \begin{align} I_{t} := \dfrac{B_t}{B_0}I_{0} \end{align} $$
where $I_t$ and $B_t$ are, respectively, the value at time $t$ of the index and the basket; similarly $I_0$ and $B_0$ are their value at the base year ($I_0$ usually is set equal to 100).
Now, the question is why this formula makes sense? A possible answer can be obtained if we remember that indices are normalized, that is, independent of a currency or put in a more general way, they are unit-free.
If an index is a unit-free number that truly represents a basket, at least we would expect that they have the same percentual changes, that is, for all $t$ we should have
$$ \begin{align} & \frac{B_t - B_0}{B_0} = \frac{I_t - I_0}{I_0} \\ \iff & \frac{B_t}{B_0} - 1 = \frac{I_t}{I_0} - 1 \\ \iff & I_{t} = \dfrac{B_t}{B_0}I_0 \end{align} $$
This is actually, the textbook definition. Thus, price indices are designed in such a way that they preserve percentual changes of their associated basket.
We now discuss the rationale behind the formula for adjusting prices for inflation. For performing this adjusment, we require an index, $I_t,$ that is related with the asset we want to adjust its price.
The textbook formula for adjusting to inflation the price $P_t$ is given by
$$ \begin{align} P_{t}^{0} := \dfrac{P_t}{I_t}I_0 \end{align} $$
Here, $P_{t}^{0}$ is the inflation-adjusted price at time time t expressed in prices for the base year, on the other hand $P_t$ is the price at time $t$ expressed in prices of that same time.
Similar to the case for the formula of an index, we could require for adjusted prices to satisfy certain natural conditions. In particular, since the asset under consideration is part of a basket, an adjusted price should preserve percentual changes of the proportion of the money value that the asset has in the basket., this means that
$$ \begin{align} & \dfrac{ \frac{P_t}{B_t} - \frac{P_0}{B_0} }{ \frac{P_0}{B_0}} = \dfrac{P_{t}^{0} - P_0}{P_0} \quad \text{ where } P_0 := P_{0}^{0} \\ \iff & \frac{P_t B_0}{B_t P_0} = \frac{P_{t}^{0}}{P_0} \\ \iff & P_{t}^{0} = P_{t} \frac{B_0}{B_t} \end{align} $$
and we know that