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3 Responses
Great question-this is exactly where the pattern of “weights that add to 1” shines! If your weights are 0.3 and 0.7, they already sum to 1, which means you’ve normalized them, so 0.3*80 + 0.7*90 = 87 is the final weighted average-no extra division needed. The general formula is (w1*x1 + w2*x2 + …)/(w1 + w2 + …); when you use 30% and 70% as decimals (0.3 and 0.7), you’ve effectively already divided by the total 100%, so the denominator is 1. If you kept the raw weights as 30 and 70, you’d do (30*80 + 70*90)/(30+70) = 8700/100 = 87, which matches. Dividing by 2 would incorrectly treat them as equally weighted, and dividing by 1.0 just leaves it the same. A nice sanity check: the answer should lean toward 90 since it has the larger weight, and 87 does. For more detail, see: https://www.mathsisfun.com/data/weighted-mean.html. Hope this helps!
No second slice of division pie needed: since 0.3 + 0.7 = 1, the weighted average is already 0.3×80 + 0.7×90 = 87 (dividing by 2 or 1 would double-count the scaling). If your weights were 30 and 70 instead, you’d do (30×80 + 70×90)/100-so you’re good here, I’m pretty sure.
I totally get that “itch” to divide at the end-I still catch myself wanting to do that sometimes. It feels like there should be one more step, right? But for weighted averages, if your weights are already percentages (or decimals) that add up to 1, then you’re done after the multiply-and-add. So 0.3*80 + 0.7*90 = 87 is already the weighted average. Dividing by 2 would actually break it.
Here’s a way I think about it in everyday terms: imagine you’re mixing a smoothie that’s 30% banana and 70% mango. The final taste should lean toward mango. In numbers, 87 is closer to 90 than to 80-perfect. If you divided by 2 afterward, you’d get 43.5, which is nowhere near either fruit’s “flavor” and clearly doesn’t make sense.
Why you don’t divide again:
– The general formula is: weighted average = (sum of weight × value) / (sum of weights).
– When your weights are 0.3 and 0.7, the sum of weights is 1. Dividing by 1 does nothing, so the multiply-and-add already gives the final answer.
– Dividing by 2 would be like pretending each score had equal weight, which ignores the 30%/70% setup.
A simple worked example with your numbers:
– Weights: 30% and 70% → decimals 0.3 and 0.7
– Values: 80 and 90
– Compute: 0.3*80 + 0.7*90 = 24 + 63 = 87
– No extra division needed, because 0.3 + 0.7 = 1
What if the weights aren’t given as percentages?
– Suppose someone said the weights are 30 and 70 (not 30% and 70%). Then you first multiply, but you do need to divide by the total weight:
– Numerator: 30*80 + 70*90 = 2400 + 6300 = 8700
– Denominator: 30 + 70 = 100
– Weighted average: 8700 / 100 = 87
– Notice how this matches the percentage version once you divide by the total weight. That’s because 30 and 70 are just un-normalized versions of 0.3 and 0.7.
Quick sanity checks I use when I’m second-guessing myself:
– Are the weights decimals or percentages that add to 1 (or 100%)? If yes, don’t divide again.
– Is the result closer to the value with the larger weight? Here, 87 is closer to 90 (which has 70% weight), so it passes the sniff test.
– If both weights were equal (like 0.5 and 0.5), the formula reduces to the usual average: 0.5*80 + 0.5*90 = 85, which is the same as (80 + 90)/2. No extra division beyond that.
So, for your case: 0.3*80 + 0.7*90 = 87, and that’s the final answer-no extra dividing by 2 or by 1.0.