Explanation:

To find the missing number in the series, we need to identify the pattern of differences between consecutive terms. \nStep 1: Calculate the differences between adjacent terms.\nDifference between $0.20$ and $0.15$: $0.20 - 0.15 = 0.05$.\nDifference between $0.30$ and $0.20$: $0.30 - 0.20 = 0.10$.\nDifference between $0.45$ and $0.30$: $0.45 - 0.30 = 0.15$.\nDifference between $0.65$ and $0.45$: $0.65 - 0.45 = 0.20$. \nStep 2: Analyze the pattern of these differences.\nThe differences are $0.05, 0.10, 0.15, 0.20$. This is an arithmetic progression where each term increases by $0.05$. \nStep 3: Determine the next difference in the pattern.\nThe next difference should be $0.20 + 0.05 = 0.25$. \nStep 4: Calculate the next term in the series.\nAdd the next difference ($0.25$) to the last given term ($0.65$): $0.65 + 0.25 = 0.90$. \nStep 5: Evaluate the options.\nOption A: $0.8$. This is incorrect.\nOption B: $0.9$. This matches our calculated result ($0.90$).\nOption C: $0.85$. This is incorrect.\nOption D: $0.95$. This is incorrect. \nTherefore, the number that replaces the question mark is $0.9$.