Step 1: Chargaff's base-pairing rule states that Adenine ($A$) pairs with Thymine ($T$), and Guanine ($G$) pairs with Cytosine ($C$). Therefore, $A=T$ and $G=C$.
Step 2: Chargaff also stated that the total number of Purines ($A+G$) always equals the total number of Pyrimidines ($T+C$).
Step 3: Therefore, $A+G = T+C$ (or $C+T$). This proves that the ratio of purines to pyrimidines is always $1:1$.