A *reserve ratio* represents a fixed ratio between the token’s total value (total supply × price) and the value of its *reserve token* balance. This ratio will be held constant by a formula as both the reserve token balance and the token’s total value (i.e. market capitalization). Since each purchase or sale of a token results in an increase or decrease of both reserve tokens and the tokens, the price of the token with respect to its reserve tokens will continuously recalculate to maintain the configured reserve ratio between the two tokens. We can calculate the dynamically changing price of the token by applying a constant reserve ratio, which is based on the Bancor system formulas and are calculated as [15]:

(1)

Where: *F* is the reserve ratio of the token, *R* is the reserve token balance, *S* is the token supply, and *P* is the token price. The reserve ratio determines the price sensitivity of the continuous token and at different reserve ratios. A higher reserve ratio between the reserve currency balance and the token will result in lower price sensitivity, meaning that each buy and sell will have a smaller than overall effect on the token’s price movement. Alternatively, a lower ratio between the reserve currency’s balance and the token will result in higher price sensitivity, meaning that each buy or sell will have a larger effect on the token’s price movement. The current price can be determined through the formula above if given *f*, *r*, and *s*:

(2)

In order to calculate price when more than one token has been purchased requires integral calculus, as the price of each incremental token purchased is different and based on the bonding curve. Therefore, we need to compute the area under the bonding curve (Figure 2) that applies to the amount of tokens purchased. From the original formula, we can calculate: the total number of reserve tokens paid, Number of tokens received, the new price of token, and new token supply. When an infinitesimal amount of tokens, *dS*, are bought, the supply of tokens increases by this amount. The price in reserve currency for these tokens is *P dS*, which are added to the reserve, meaning *dR = P dS*. Also, since *R = FSP*, we have *dR = d(FSP) = Fd(SP)* = *F(S dP + P dS)*, which can be solved to:

(3)

(4) above provides the price of the new token given a change in supply. If a user is buying more than one new token, T, then we can calculate the total amount paid, E, by applying integration as follows:

(4)

With the value for E, the total amount paid in reserve currency units, we can solve for T, which is the number of new tokens given E:

(5)

The BRC protocol will allow multiple reserve currencies that can be used to purchase BRC tokens. If there are m different reserve currencies, then for each currency *i* *∈ {*1,2,…,*m}* we have an amount in reserve R_{i}, fractional reserve ratio F_{i}, and price P_{i}. Buying and selling tokens with one reserve currency *i* will affect the price of the BRC token while maintaining the same supply of the other reserve currencies. Given that there are R_{0} outstanding tokens and reserve r*i*_{0} of each currency *i*, we can solve for the supply of tokens s given multiple reserve currencies and the amount of T tokens received given R1, R2, R3,…m tokens paid can be solved as: [16]

(6)

(7)