The first method is dollar balancing and the second one is beta balancing.

For the first case: Dollar balancing

Let's say you have $1000 worth of stock A. Simply divide this with the price of B. The number you get is the number you need to short B to dollar balance your portfolio. The dollar balancing is a rough way to hedge a position.

For the second case: Beta balancing

Let's say you long 100 shares of stock A. You need to short 100 * hedge ratio(beta) to rebalance your portfolio.

The first method is dollar neutral and the second one is based on the relationship of price movement between two assets.

For the first method of dollar neutral

Let's say you want to keep the amount invested in stock A and stock B same. Then, simply divide $1000 with the price of A and B. The number you get is the number of shares of A and B you need to buy/sell to make the pair dollar neutral.

For the second method, you need to find the relationship between two stocks A and B. Use that to calculate the spread. For example, the spread can be formed as 1 * stock A - slope * stock B. Where the slope of the line resulting from regressing A and b prices becomes the number of shares of stock b to buy for every 1 share of stock A.