How do I properly calculate a weighted average when the weights don't add up to 1?

I've been trying to work through a problem involving weighted averages and I keep secondguessing myself on the normalization step.

Say I have a set of values like 80, 90, and 70, with corresponding weights of 3, 5, and 2. The standard approach is to multiply each value by its weight, sum those products, then divide by the total sum of the weights. So it would be (80×3 + 90×5 + 70×2) divided by (3+5+2), giving (240 + 450 + 140) / 10 = 83.That part feels straightforward when the weights sum to a clean number like 10. But what if the weights were something like 1.5, 2.3, and 0.8? Do I still just divide by their sum (4.6 in this case), or do I need to first convert them into proper proportions that sum to 1 before doing anything?I tried both approaches and got the same answer, which makes me think they're equivalent, but I'm not 100% sure why that is mathematically. Is it just because dividing by the total sum effectively normalizes them regardless? Or am I missing a case where the two approaches would actually diverge?I feel like I understand the mechanics but not the deeper reason it works. Would appreciate any explanation or intuition behind this.